How do you integrate #int [s(s+6)] / [(s+3)(s^2+6s+18)]# using partial fractions?
The answer is
Now let's break it down into partial fractions.
So
Consequently,
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To integrate theTo integrate the givenTo integrate the given expressionTo integrate the given expression usingTo integrate (To integrate the given expression using partialTo integrate ( \To integrate the given expression using partial fractionsTo integrate ( \fracTo integrate the given expression using partial fractions,To integrate ( \frac{sTo integrate the given expression using partial fractions, firstTo integrate ( \frac{s(sTo integrate the given expression using partial fractions, first factorizeTo integrate ( \frac{s(s+To integrate the given expression using partial fractions, first factorize theTo integrate ( \frac{s(s+6To integrate the given expression using partial fractions, first factorize the denominatorTo integrate ( \frac{s(s+6)}To integrate the given expression using partial fractions, first factorize the denominator:To integrate ( \frac{s(s+6)}{(To integrate the given expression using partial fractions, first factorize the denominator: (To integrate ( \frac{s(s+6)}{(sTo integrate the given expression using partial fractions, first factorize the denominator: ( (To integrate ( \frac{s(s+6)}{(s+To integrate the given expression using partial fractions, first factorize the denominator: ( (sTo integrate ( \frac{s(s+6)}{(s+3To integrate the given expression using partial fractions, first factorize the denominator: ( (s+To integrate ( \frac{s(s+6)}{(s+3)(To integrate the given expression using partial fractions, first factorize the denominator: ( (s+3To integrate ( \frac{s(s+6)}{(s+3)(sTo integrate the given expression using partial fractions, first factorize the denominator: ( (s+3)(To integrate ( \frac{s(s+6)}{(s+3)(s^To integrate the given expression using partial fractions, first factorize the denominator: ( (s+3)(sTo integrate ( \frac{s(s+6)}{(s+3)(s^2To integrate the given expression using partial fractions, first factorize the denominator: ( (s+3)(s^2To integrate ( \frac{s(s+6)}{(s+3)(s^2+To integrate the given expression using partial fractions, first factorize the denominator: ( (s+3)(s^2+To integrate ( \frac{s(s+6)}{(s+3)(s^2+6sTo integrate the given expression using partial fractions, first factorize the denominator: ( (s+3)(s^2+6sTo integrate ( \frac{s(s+6)}{(s+3)(s^2+6s+To integrate the given expression using partial fractions, first factorize the denominator: ( (s+3)(s^2+6s+To integrate ( \frac{s(s+6)}{(s+3)(s^2+6s+18To integrate the given expression using partial fractions, first factorize the denominator: ( (s+3)(s^2+6s+18) =To integrate ( \frac{s(s+6)}{(s+3)(s^2+6s+18)}To integrate the given expression using partial fractions, first factorize the denominator: ( (s+3)(s^2+6s+18) = (To integrate ( \frac{s(s+6)}{(s+3)(s^2+6s+18)} \To integrate the given expression using partial fractions, first factorize the denominator: ( (s+3)(s^2+6s+18) = (s+3)(To integrate ( \frac{s(s+6)}{(s+3)(s^2+6s+18)} )To integrate the given expression using partial fractions, first factorize the denominator: ( (s+3)(s^2+6s+18) = (s+3)(s+3To integrate ( \frac{s(s+6)}{(s+3)(s^2+6s+18)} ) usingTo integrate the given expression using partial fractions, first factorize the denominator: ( (s+3)(s^2+6s+18) = (s+3)(s+3+To integrate ( \frac{s(s+6)}{(s+3)(s^2+6s+18)} ) using partialTo integrate the given expression using partial fractions, first factorize the denominator: ( (s+3)(s^2+6s+18) = (s+3)(s+3+3To integrate ( \frac{s(s+6)}{(s+3)(s^2+6s+18)} ) using partial fractionsTo integrate the given expression using partial fractions, first factorize the denominator: ( (s+3)(s^2+6s+18) = (s+3)(s+3+3iTo integrate ( \frac{s(s+6)}{(s+3)(s^2+6s+18)} ) using partial fractions,To integrate the given expression using partial fractions, first factorize the denominator: ( (s+3)(s^2+6s+18) = (s+3)(s+3+3i)(To integrate ( \frac{s(s+6)}{(s+3)(s^2+6s+18)} ) using partial fractions, followTo integrate the given expression using partial fractions, first factorize the denominator: ( (s+3)(s^2+6s+18) = (s+3)(s+3+3i)(sTo integrate ( \frac{s(s+6)}{(s+3)(s^2+6s+18)} ) using partial fractions, follow theseTo integrate the given expression using partial fractions, first factorize the denominator: ( (s+3)(s^2+6s+18) = (s+3)(s+3+3i)(s+To integrate ( \frac{s(s+6)}{(s+3)(s^2+6s+18)} ) using partial fractions, follow these stepsTo integrate the given expression using partial fractions, first factorize the denominator: ( (s+3)(s^2+6s+18) = (s+3)(s+3+3i)(s+33To integrate ( \frac{s(s+6)}{(s+3)(s^2+6s+18)} ) using partial fractions, follow these steps:
To integrate the given expression using partial fractions, first factorize the denominator: ( (s+3)(s^2+6s+18) = (s+3)(s+3+3i)(s+33i)To integrate ( \frac{s(s+6)}{(s+3)(s^2+6s+18)} ) using partial fractions, follow these steps:
1.To integrate the given expression using partial fractions, first factorize the denominator: ( (s+3)(s^2+6s+18) = (s+3)(s+3+3i)(s+33i) \To integrate ( \frac{s(s+6)}{(s+3)(s^2+6s+18)} ) using partial fractions, follow these steps:

FactorTo integrate the given expression using partial fractions, first factorize the denominator: ( (s+3)(s^2+6s+18) = (s+3)(s+3+3i)(s+33i) ).To integrate ( \frac{s(s+6)}{(s+3)(s^2+6s+18)} ) using partial fractions, follow these steps:

Factor the denominatorTo integrate the given expression using partial fractions, first factorize the denominator: ( (s+3)(s^2+6s+18) = (s+3)(s+3+3i)(s+33i) ). ThenTo integrate ( \frac{s(s+6)}{(s+3)(s^2+6s+18)} ) using partial fractions, follow these steps:

Factor the denominator (To integrate the given expression using partial fractions, first factorize the denominator: ( (s+3)(s^2+6s+18) = (s+3)(s+3+3i)(s+33i) ). Then,To integrate ( \frac{s(s+6)}{(s+3)(s^2+6s+18)} ) using partial fractions, follow these steps:

Factor the denominator ( (To integrate the given expression using partial fractions, first factorize the denominator: ( (s+3)(s^2+6s+18) = (s+3)(s+3+3i)(s+33i) ). Then, expressTo integrate ( \frac{s(s+6)}{(s+3)(s^2+6s+18)} ) using partial fractions, follow these steps:

Factor the denominator ( (sTo integrate the given expression using partial fractions, first factorize the denominator: ( (s+3)(s^2+6s+18) = (s+3)(s+3+3i)(s+33i) ). Then, express theTo integrate ( \frac{s(s+6)}{(s+3)(s^2+6s+18)} ) using partial fractions, follow these steps:

Factor the denominator ( (s+To integrate the given expression using partial fractions, first factorize the denominator: ( (s+3)(s^2+6s+18) = (s+3)(s+3+3i)(s+33i) ). Then, express the givenTo integrate ( \frac{s(s+6)}{(s+3)(s^2+6s+18)} ) using partial fractions, follow these steps:

Factor the denominator ( (s+3)(To integrate the given expression using partial fractions, first factorize the denominator: ( (s+3)(s^2+6s+18) = (s+3)(s+3+3i)(s+33i) ). Then, express the given fractionTo integrate ( \frac{s(s+6)}{(s+3)(s^2+6s+18)} ) using partial fractions, follow these steps:

Factor the denominator ( (s+3)(s^2To integrate the given expression using partial fractions, first factorize the denominator: ( (s+3)(s^2+6s+18) = (s+3)(s+3+3i)(s+33i) ). Then, express the given fraction asTo integrate ( \frac{s(s+6)}{(s+3)(s^2+6s+18)} ) using partial fractions, follow these steps:

Factor the denominator ( (s+3)(s^2+6To integrate the given expression using partial fractions, first factorize the denominator: ( (s+3)(s^2+6s+18) = (s+3)(s+3+3i)(s+33i) ). Then, express the given fraction as theTo integrate ( \frac{s(s+6)}{(s+3)(s^2+6s+18)} ) using partial fractions, follow these steps:

Factor the denominator ( (s+3)(s^2+6s+18)To integrate the given expression using partial fractions, first factorize the denominator: ( (s+3)(s^2+6s+18) = (s+3)(s+3+3i)(s+33i) ). Then, express the given fraction as the sumTo integrate ( \frac{s(s+6)}{(s+3)(s^2+6s+18)} ) using partial fractions, follow these steps:

Factor the denominator ( (s+3)(s^2+6s+18) \To integrate the given expression using partial fractions, first factorize the denominator: ( (s+3)(s^2+6s+18) = (s+3)(s+3+3i)(s+33i) ). Then, express the given fraction as the sum ofTo integrate ( \frac{s(s+6)}{(s+3)(s^2+6s+18)} ) using partial fractions, follow these steps:

Factor the denominator ( (s+3)(s^2+6s+18) ). To integrate the given expression using partial fractions, first factorize the denominator: ( (s+3)(s^2+6s+18) = (s+3)(s+3+3i)(s+33i) ). Then, express the given fraction as the sum of twoTo integrate ( \frac{s(s+6)}{(s+3)(s^2+6s+18)} ) using partial fractions, follow these steps:

Factor the denominator ( (s+3)(s^2+6s+18) ). 2To integrate the given expression using partial fractions, first factorize the denominator: ( (s+3)(s^2+6s+18) = (s+3)(s+3+3i)(s+33i) ). Then, express the given fraction as the sum of two orTo integrate ( \frac{s(s+6)}{(s+3)(s^2+6s+18)} ) using partial fractions, follow these steps:

Factor the denominator ( (s+3)(s^2+6s+18) ). 2.To integrate the given expression using partial fractions, first factorize the denominator: ( (s+3)(s^2+6s+18) = (s+3)(s+3+3i)(s+33i) ). Then, express the given fraction as the sum of two or threeTo integrate ( \frac{s(s+6)}{(s+3)(s^2+6s+18)} ) using partial fractions, follow these steps:

Factor the denominator ( (s+3)(s^2+6s+18) ).

ExpressTo integrate the given expression using partial fractions, first factorize the denominator: ( (s+3)(s^2+6s+18) = (s+3)(s+3+3i)(s+33i) ). Then, express the given fraction as the sum of two or three partialTo integrate ( \frac{s(s+6)}{(s+3)(s^2+6s+18)} ) using partial fractions, follow these steps:

Factor the denominator ( (s+3)(s^2+6s+18) ).

Express theTo integrate the given expression using partial fractions, first factorize the denominator: ( (s+3)(s^2+6s+18) = (s+3)(s+3+3i)(s+33i) ). Then, express the given fraction as the sum of two or three partial fractionsTo integrate ( \frac{s(s+6)}{(s+3)(s^2+6s+18)} ) using partial fractions, follow these steps:

Factor the denominator ( (s+3)(s^2+6s+18) ).

Express the fractionTo integrate the given expression using partial fractions, first factorize the denominator: ( (s+3)(s^2+6s+18) = (s+3)(s+3+3i)(s+33i) ). Then, express the given fraction as the sum of two or three partial fractions.To integrate ( \frac{s(s+6)}{(s+3)(s^2+6s+18)} ) using partial fractions, follow these steps:

Factor the denominator ( (s+3)(s^2+6s+18) ).

Express the fraction asTo integrate the given expression using partial fractions, first factorize the denominator: ( (s+3)(s^2+6s+18) = (s+3)(s+3+3i)(s+33i) ). Then, express the given fraction as the sum of two or three partial fractions. TheTo integrate ( \frac{s(s+6)}{(s+3)(s^2+6s+18)} ) using partial fractions, follow these steps:

Factor the denominator ( (s+3)(s^2+6s+18) ).

Express the fraction as aTo integrate the given expression using partial fractions, first factorize the denominator: ( (s+3)(s^2+6s+18) = (s+3)(s+3+3i)(s+33i) ). Then, express the given fraction as the sum of two or three partial fractions. The generalTo integrate ( \frac{s(s+6)}{(s+3)(s^2+6s+18)} ) using partial fractions, follow these steps:

Factor the denominator ( (s+3)(s^2+6s+18) ).

Express the fraction as a sumTo integrate the given expression using partial fractions, first factorize the denominator: ( (s+3)(s^2+6s+18) = (s+3)(s+3+3i)(s+33i) ). Then, express the given fraction as the sum of two or three partial fractions. The general formTo integrate ( \frac{s(s+6)}{(s+3)(s^2+6s+18)} ) using partial fractions, follow these steps:

Factor the denominator ( (s+3)(s^2+6s+18) ).

Express the fraction as a sum ofTo integrate the given expression using partial fractions, first factorize the denominator: ( (s+3)(s^2+6s+18) = (s+3)(s+3+3i)(s+33i) ). Then, express the given fraction as the sum of two or three partial fractions. The general form forTo integrate ( \frac{s(s+6)}{(s+3)(s^2+6s+18)} ) using partial fractions, follow these steps:

Factor the denominator ( (s+3)(s^2+6s+18) ).

Express the fraction as a sum of partialTo integrate the given expression using partial fractions, first factorize the denominator: ( (s+3)(s^2+6s+18) = (s+3)(s+3+3i)(s+33i) ). Then, express the given fraction as the sum of two or three partial fractions. The general form for partialTo integrate ( \frac{s(s+6)}{(s+3)(s^2+6s+18)} ) using partial fractions, follow these steps:

Factor the denominator ( (s+3)(s^2+6s+18) ).

Express the fraction as a sum of partial fractionsTo integrate the given expression using partial fractions, first factorize the denominator: ( (s+3)(s^2+6s+18) = (s+3)(s+3+3i)(s+33i) ). Then, express the given fraction as the sum of two or three partial fractions. The general form for partial fraction decompositionTo integrate ( \frac{s(s+6)}{(s+3)(s^2+6s+18)} ) using partial fractions, follow these steps:

Factor the denominator ( (s+3)(s^2+6s+18) ).

Express the fraction as a sum of partial fractions. To integrate the given expression using partial fractions, first factorize the denominator: ( (s+3)(s^2+6s+18) = (s+3)(s+3+3i)(s+33i) ). Then, express the given fraction as the sum of two or three partial fractions. The general form for partial fraction decomposition isTo integrate ( \frac{s(s+6)}{(s+3)(s^2+6s+18)} ) using partial fractions, follow these steps:

Factor the denominator ( (s+3)(s^2+6s+18) ).

Express the fraction as a sum of partial fractions. 3To integrate the given expression using partial fractions, first factorize the denominator: ( (s+3)(s^2+6s+18) = (s+3)(s+3+3i)(s+33i) ). Then, express the given fraction as the sum of two or three partial fractions. The general form for partial fraction decomposition is:
To integrate ( \frac{s(s+6)}{(s+3)(s^2+6s+18)} ) using partial fractions, follow these steps:
 Factor the denominator ( (s+3)(s^2+6s+18) ).
 Express the fraction as a sum of partial fractions. 3.To integrate the given expression using partial fractions, first factorize the denominator: ( (s+3)(s^2+6s+18) = (s+3)(s+3+3i)(s+33i) ). Then, express the given fraction as the sum of two or three partial fractions. The general form for partial fraction decomposition is:
\To integrate ( \frac{s(s+6)}{(s+3)(s^2+6s+18)} ) using partial fractions, follow these steps:
 Factor the denominator ( (s+3)(s^2+6s+18) ).
 Express the fraction as a sum of partial fractions.
 Solve forTo integrate the given expression using partial fractions, first factorize the denominator: ( (s+3)(s^2+6s+18) = (s+3)(s+3+3i)(s+33i) ). Then, express the given fraction as the sum of two or three partial fractions. The general form for partial fraction decomposition is:
[ To integrate ( \frac{s(s+6)}{(s+3)(s^2+6s+18)} ) using partial fractions, follow these steps:
 Factor the denominator ( (s+3)(s^2+6s+18) ).
 Express the fraction as a sum of partial fractions.
 Solve for theTo integrate the given expression using partial fractions, first factorize the denominator: ( (s+3)(s^2+6s+18) = (s+3)(s+3+3i)(s+33i) ). Then, express the given fraction as the sum of two or three partial fractions. The general form for partial fraction decomposition is:
[ \To integrate ( \frac{s(s+6)}{(s+3)(s^2+6s+18)} ) using partial fractions, follow these steps:
 Factor the denominator ( (s+3)(s^2+6s+18) ).
 Express the fraction as a sum of partial fractions.
 Solve for the unknownTo integrate the given expression using partial fractions, first factorize the denominator: ( (s+3)(s^2+6s+18) = (s+3)(s+3+3i)(s+33i) ). Then, express the given fraction as the sum of two or three partial fractions. The general form for partial fraction decomposition is:
[ \fracTo integrate ( \frac{s(s+6)}{(s+3)(s^2+6s+18)} ) using partial fractions, follow these steps:
 Factor the denominator ( (s+3)(s^2+6s+18) ).
 Express the fraction as a sum of partial fractions.
 Solve for the unknown coefficientsTo integrate the given expression using partial fractions, first factorize the denominator: ( (s+3)(s^2+6s+18) = (s+3)(s+3+3i)(s+33i) ). Then, express the given fraction as the sum of two or three partial fractions. The general form for partial fraction decomposition is:
[ \frac{A}{To integrate ( \frac{s(s+6)}{(s+3)(s^2+6s+18)} ) using partial fractions, follow these steps:
 Factor the denominator ( (s+3)(s^2+6s+18) ).
 Express the fraction as a sum of partial fractions.
 Solve for the unknown coefficients. To integrate the given expression using partial fractions, first factorize the denominator: ( (s+3)(s^2+6s+18) = (s+3)(s+3+3i)(s+33i) ). Then, express the given fraction as the sum of two or three partial fractions. The general form for partial fraction decomposition is:
[ \frac{A}{s+3To integrate ( \frac{s(s+6)}{(s+3)(s^2+6s+18)} ) using partial fractions, follow these steps:
 Factor the denominator ( (s+3)(s^2+6s+18) ).
 Express the fraction as a sum of partial fractions.
 Solve for the unknown coefficients. 4To integrate the given expression using partial fractions, first factorize the denominator: ( (s+3)(s^2+6s+18) = (s+3)(s+3+3i)(s+33i) ). Then, express the given fraction as the sum of two or three partial fractions. The general form for partial fraction decomposition is:
[ \frac{A}{s+3}To integrate ( \frac{s(s+6)}{(s+3)(s^2+6s+18)} ) using partial fractions, follow these steps:
 Factor the denominator ( (s+3)(s^2+6s+18) ).
 Express the fraction as a sum of partial fractions.
 Solve for the unknown coefficients. 4.To integrate the given expression using partial fractions, first factorize the denominator: ( (s+3)(s^2+6s+18) = (s+3)(s+3+3i)(s+33i) ). Then, express the given fraction as the sum of two or three partial fractions. The general form for partial fraction decomposition is:
[ \frac{A}{s+3} +To integrate ( \frac{s(s+6)}{(s+3)(s^2+6s+18)} ) using partial fractions, follow these steps:
 Factor the denominator ( (s+3)(s^2+6s+18) ).
 Express the fraction as a sum of partial fractions.
 Solve for the unknown coefficients.
 IntTo integrate the given expression using partial fractions, first factorize the denominator: ( (s+3)(s^2+6s+18) = (s+3)(s+3+3i)(s+33i) ). Then, express the given fraction as the sum of two or three partial fractions. The general form for partial fraction decomposition is:
[ \frac{A}{s+3} + \To integrate ( \frac{s(s+6)}{(s+3)(s^2+6s+18)} ) using partial fractions, follow these steps:
 Factor the denominator ( (s+3)(s^2+6s+18) ).
 Express the fraction as a sum of partial fractions.
 Solve for the unknown coefficients.
 IntegrateTo integrate the given expression using partial fractions, first factorize the denominator: ( (s+3)(s^2+6s+18) = (s+3)(s+3+3i)(s+33i) ). Then, express the given fraction as the sum of two or three partial fractions. The general form for partial fraction decomposition is:
[ \frac{A}{s+3} + \fracTo integrate ( \frac{s(s+6)}{(s+3)(s^2+6s+18)} ) using partial fractions, follow these steps:
 Factor the denominator ( (s+3)(s^2+6s+18) ).
 Express the fraction as a sum of partial fractions.
 Solve for the unknown coefficients.
 Integrate eachTo integrate the given expression using partial fractions, first factorize the denominator: ( (s+3)(s^2+6s+18) = (s+3)(s+3+3i)(s+33i) ). Then, express the given fraction as the sum of two or three partial fractions. The general form for partial fraction decomposition is:
[ \frac{A}{s+3} + \frac{To integrate ( \frac{s(s+6)}{(s+3)(s^2+6s+18)} ) using partial fractions, follow these steps:
 Factor the denominator ( (s+3)(s^2+6s+18) ).
 Express the fraction as a sum of partial fractions.
 Solve for the unknown coefficients.
 Integrate each termTo integrate the given expression using partial fractions, first factorize the denominator: ( (s+3)(s^2+6s+18) = (s+3)(s+3+3i)(s+33i) ). Then, express the given fraction as the sum of two or three partial fractions. The general form for partial fraction decomposition is:
[ \frac{A}{s+3} + \frac{Bs +To integrate ( \frac{s(s+6)}{(s+3)(s^2+6s+18)} ) using partial fractions, follow these steps:
 Factor the denominator ( (s+3)(s^2+6s+18) ).
 Express the fraction as a sum of partial fractions.
 Solve for the unknown coefficients.
 Integrate each term separatelyTo integrate the given expression using partial fractions, first factorize the denominator: ( (s+3)(s^2+6s+18) = (s+3)(s+3+3i)(s+33i) ). Then, express the given fraction as the sum of two or three partial fractions. The general form for partial fraction decomposition is:
[ \frac{A}{s+3} + \frac{Bs + CTo integrate ( \frac{s(s+6)}{(s+3)(s^2+6s+18)} ) using partial fractions, follow these steps:
 Factor the denominator ( (s+3)(s^2+6s+18) ).
 Express the fraction as a sum of partial fractions.
 Solve for the unknown coefficients.
 Integrate each term separately. To integrate the given expression using partial fractions, first factorize the denominator: ( (s+3)(s^2+6s+18) = (s+3)(s+3+3i)(s+33i) ). Then, express the given fraction as the sum of two or three partial fractions. The general form for partial fraction decomposition is:
[ \frac{A}{s+3} + \frac{Bs + C}{To integrate ( \frac{s(s+6)}{(s+3)(s^2+6s+18)} ) using partial fractions, follow these steps:
 Factor the denominator ( (s+3)(s^2+6s+18) ).
 Express the fraction as a sum of partial fractions.
 Solve for the unknown coefficients.
 Integrate each term separately. 5To integrate the given expression using partial fractions, first factorize the denominator: ( (s+3)(s^2+6s+18) = (s+3)(s+3+3i)(s+33i) ). Then, express the given fraction as the sum of two or three partial fractions. The general form for partial fraction decomposition is:
[ \frac{A}{s+3} + \frac{Bs + C}{sTo integrate ( \frac{s(s+6)}{(s+3)(s^2+6s+18)} ) using partial fractions, follow these steps:
 Factor the denominator ( (s+3)(s^2+6s+18) ).
 Express the fraction as a sum of partial fractions.
 Solve for the unknown coefficients.
 Integrate each term separately. 5.To integrate the given expression using partial fractions, first factorize the denominator: ( (s+3)(s^2+6s+18) = (s+3)(s+3+3i)(s+33i) ). Then, express the given fraction as the sum of two or three partial fractions. The general form for partial fraction decomposition is:
[ \frac{A}{s+3} + \frac{Bs + C}{s^To integrate ( \frac{s(s+6)}{(s+3)(s^2+6s+18)} ) using partial fractions, follow these steps:
 Factor the denominator ( (s+3)(s^2+6s+18) ).
 Express the fraction as a sum of partial fractions.
 Solve for the unknown coefficients.
 Integrate each term separately.
 CombineTo integrate the given expression using partial fractions, first factorize the denominator: ( (s+3)(s^2+6s+18) = (s+3)(s+3+3i)(s+33i) ). Then, express the given fraction as the sum of two or three partial fractions. The general form for partial fraction decomposition is:
[ \frac{A}{s+3} + \frac{Bs + C}{s^2To integrate ( \frac{s(s+6)}{(s+3)(s^2+6s+18)} ) using partial fractions, follow these steps:
 Factor the denominator ( (s+3)(s^2+6s+18) ).
 Express the fraction as a sum of partial fractions.
 Solve for the unknown coefficients.
 Integrate each term separately.
 Combine theTo integrate the given expression using partial fractions, first factorize the denominator: ( (s+3)(s^2+6s+18) = (s+3)(s+3+3i)(s+33i) ). Then, express the given fraction as the sum of two or three partial fractions. The general form for partial fraction decomposition is:
[ \frac{A}{s+3} + \frac{Bs + C}{s^2+To integrate ( \frac{s(s+6)}{(s+3)(s^2+6s+18)} ) using partial fractions, follow these steps:
 Factor the denominator ( (s+3)(s^2+6s+18) ).
 Express the fraction as a sum of partial fractions.
 Solve for the unknown coefficients.
 Integrate each term separately.
 Combine the results.
To integrate the given expression using partial fractions, first factorize the denominator: ( (s+3)(s^2+6s+18) = (s+3)(s+3+3i)(s+33i) ). Then, express the given fraction as the sum of two or three partial fractions. The general form for partial fraction decomposition is:
[ \frac{A}{s+3} + \frac{Bs + C}{s^2+6sTo integrate ( \frac{s(s+6)}{(s+3)(s^2+6s+18)} ) using partial fractions, follow these steps:
 Factor the denominator ( (s+3)(s^2+6s+18) ).
 Express the fraction as a sum of partial fractions.
 Solve for the unknown coefficients.
 Integrate each term separately.
 Combine the results.
StepTo integrate the given expression using partial fractions, first factorize the denominator: ( (s+3)(s^2+6s+18) = (s+3)(s+3+3i)(s+33i) ). Then, express the given fraction as the sum of two or three partial fractions. The general form for partial fraction decomposition is:
[ \frac{A}{s+3} + \frac{Bs + C}{s^2+6s+To integrate ( \frac{s(s+6)}{(s+3)(s^2+6s+18)} ) using partial fractions, follow these steps:
 Factor the denominator ( (s+3)(s^2+6s+18) ).
 Express the fraction as a sum of partial fractions.
 Solve for the unknown coefficients.
 Integrate each term separately.
 Combine the results.
Step 1To integrate the given expression using partial fractions, first factorize the denominator: ( (s+3)(s^2+6s+18) = (s+3)(s+3+3i)(s+33i) ). Then, express the given fraction as the sum of two or three partial fractions. The general form for partial fraction decomposition is:
[ \frac{A}{s+3} + \frac{Bs + C}{s^2+6s+18To integrate ( \frac{s(s+6)}{(s+3)(s^2+6s+18)} ) using partial fractions, follow these steps:
 Factor the denominator ( (s+3)(s^2+6s+18) ).
 Express the fraction as a sum of partial fractions.
 Solve for the unknown coefficients.
 Integrate each term separately.
 Combine the results.
Step 1: FactorTo integrate the given expression using partial fractions, first factorize the denominator: ( (s+3)(s^2+6s+18) = (s+3)(s+3+3i)(s+33i) ). Then, express the given fraction as the sum of two or three partial fractions. The general form for partial fraction decomposition is:
[ \frac{A}{s+3} + \frac{Bs + C}{s^2+6s+18} ]
To integrate ( \frac{s(s+6)}{(s+3)(s^2+6s+18)} ) using partial fractions, follow these steps:
 Factor the denominator ( (s+3)(s^2+6s+18) ).
 Express the fraction as a sum of partial fractions.
 Solve for the unknown coefficients.
 Integrate each term separately.
 Combine the results.
Step 1: Factor the denominatorTo integrate the given expression using partial fractions, first factorize the denominator: ( (s+3)(s^2+6s+18) = (s+3)(s+3+3i)(s+33i) ). Then, express the given fraction as the sum of two or three partial fractions. The general form for partial fraction decomposition is:
[ \frac{A}{s+3} + \frac{Bs + C}{s^2+6s+18} ]
Solve forTo integrate ( \frac{s(s+6)}{(s+3)(s^2+6s+18)} ) using partial fractions, follow these steps:
 Factor the denominator ( (s+3)(s^2+6s+18) ).
 Express the fraction as a sum of partial fractions.
 Solve for the unknown coefficients.
 Integrate each term separately.
 Combine the results.
Step 1: Factor the denominator (To integrate the given expression using partial fractions, first factorize the denominator: ( (s+3)(s^2+6s+18) = (s+3)(s+3+3i)(s+33i) ). Then, express the given fraction as the sum of two or three partial fractions. The general form for partial fraction decomposition is:
[ \frac{A}{s+3} + \frac{Bs + C}{s^2+6s+18} ]
Solve for theTo integrate ( \frac{s(s+6)}{(s+3)(s^2+6s+18)} ) using partial fractions, follow these steps:
 Factor the denominator ( (s+3)(s^2+6s+18) ).
 Express the fraction as a sum of partial fractions.
 Solve for the unknown coefficients.
 Integrate each term separately.
 Combine the results.
Step 1: Factor the denominator ( (sTo integrate the given expression using partial fractions, first factorize the denominator: ( (s+3)(s^2+6s+18) = (s+3)(s+3+3i)(s+33i) ). Then, express the given fraction as the sum of two or three partial fractions. The general form for partial fraction decomposition is:
[ \frac{A}{s+3} + \frac{Bs + C}{s^2+6s+18} ]
Solve for the constantsTo integrate ( \frac{s(s+6)}{(s+3)(s^2+6s+18)} ) using partial fractions, follow these steps:
 Factor the denominator ( (s+3)(s^2+6s+18) ).
 Express the fraction as a sum of partial fractions.
 Solve for the unknown coefficients.
 Integrate each term separately.
 Combine the results.
Step 1: Factor the denominator ( (s+To integrate the given expression using partial fractions, first factorize the denominator: ( (s+3)(s^2+6s+18) = (s+3)(s+3+3i)(s+33i) ). Then, express the given fraction as the sum of two or three partial fractions. The general form for partial fraction decomposition is:
[ \frac{A}{s+3} + \frac{Bs + C}{s^2+6s+18} ]
Solve for the constants (To integrate ( \frac{s(s+6)}{(s+3)(s^2+6s+18)} ) using partial fractions, follow these steps:
 Factor the denominator ( (s+3)(s^2+6s+18) ).
 Express the fraction as a sum of partial fractions.
 Solve for the unknown coefficients.
 Integrate each term separately.
 Combine the results.
Step 1: Factor the denominator ( (s+3To integrate the given expression using partial fractions, first factorize the denominator: ( (s+3)(s^2+6s+18) = (s+3)(s+3+3i)(s+33i) ). Then, express the given fraction as the sum of two or three partial fractions. The general form for partial fraction decomposition is:
[ \frac{A}{s+3} + \frac{Bs + C}{s^2+6s+18} ]
Solve for the constants ( ATo integrate ( \frac{s(s+6)}{(s+3)(s^2+6s+18)} ) using partial fractions, follow these steps:
 Factor the denominator ( (s+3)(s^2+6s+18) ).
 Express the fraction as a sum of partial fractions.
 Solve for the unknown coefficients.
 Integrate each term separately.
 Combine the results.
Step 1: Factor the denominator ( (s+3)(To integrate the given expression using partial fractions, first factorize the denominator: ( (s+3)(s^2+6s+18) = (s+3)(s+3+3i)(s+33i) ). Then, express the given fraction as the sum of two or three partial fractions. The general form for partial fraction decomposition is:
[ \frac{A}{s+3} + \frac{Bs + C}{s^2+6s+18} ]
Solve for the constants ( A \To integrate ( \frac{s(s+6)}{(s+3)(s^2+6s+18)} ) using partial fractions, follow these steps:
 Factor the denominator ( (s+3)(s^2+6s+18) ).
 Express the fraction as a sum of partial fractions.
 Solve for the unknown coefficients.
 Integrate each term separately.
 Combine the results.
Step 1: Factor the denominator ( (s+3)(sTo integrate the given expression using partial fractions, first factorize the denominator: ( (s+3)(s^2+6s+18) = (s+3)(s+3+3i)(s+33i) ). Then, express the given fraction as the sum of two or three partial fractions. The general form for partial fraction decomposition is:
[ \frac{A}{s+3} + \frac{Bs + C}{s^2+6s+18} ]
Solve for the constants ( A ),To integrate ( \frac{s(s+6)}{(s+3)(s^2+6s+18)} ) using partial fractions, follow these steps:
 Factor the denominator ( (s+3)(s^2+6s+18) ).
 Express the fraction as a sum of partial fractions.
 Solve for the unknown coefficients.
 Integrate each term separately.
 Combine the results.
Step 1: Factor the denominator ( (s+3)(s^To integrate the given expression using partial fractions, first factorize the denominator: ( (s+3)(s^2+6s+18) = (s+3)(s+3+3i)(s+33i) ). Then, express the given fraction as the sum of two or three partial fractions. The general form for partial fraction decomposition is:
[ \frac{A}{s+3} + \frac{Bs + C}{s^2+6s+18} ]
Solve for the constants ( A ), (To integrate ( \frac{s(s+6)}{(s+3)(s^2+6s+18)} ) using partial fractions, follow these steps:
 Factor the denominator ( (s+3)(s^2+6s+18) ).
 Express the fraction as a sum of partial fractions.
 Solve for the unknown coefficients.
 Integrate each term separately.
 Combine the results.
Step 1: Factor the denominator ( (s+3)(s^2To integrate the given expression using partial fractions, first factorize the denominator: ( (s+3)(s^2+6s+18) = (s+3)(s+3+3i)(s+33i) ). Then, express the given fraction as the sum of two or three partial fractions. The general form for partial fraction decomposition is:
[ \frac{A}{s+3} + \frac{Bs + C}{s^2+6s+18} ]
Solve for the constants ( A ), ( BTo integrate ( \frac{s(s+6)}{(s+3)(s^2+6s+18)} ) using partial fractions, follow these steps:
 Factor the denominator ( (s+3)(s^2+6s+18) ).
 Express the fraction as a sum of partial fractions.
 Solve for the unknown coefficients.
 Integrate each term separately.
 Combine the results.
Step 1: Factor the denominator ( (s+3)(s^2+6To integrate the given expression using partial fractions, first factorize the denominator: ( (s+3)(s^2+6s+18) = (s+3)(s+3+3i)(s+33i) ). Then, express the given fraction as the sum of two or three partial fractions. The general form for partial fraction decomposition is:
[ \frac{A}{s+3} + \frac{Bs + C}{s^2+6s+18} ]
Solve for the constants ( A ), ( B \To integrate ( \frac{s(s+6)}{(s+3)(s^2+6s+18)} ) using partial fractions, follow these steps:
 Factor the denominator ( (s+3)(s^2+6s+18) ).
 Express the fraction as a sum of partial fractions.
 Solve for the unknown coefficients.
 Integrate each term separately.
 Combine the results.
Step 1: Factor the denominator ( (s+3)(s^2+6sTo integrate the given expression using partial fractions, first factorize the denominator: ( (s+3)(s^2+6s+18) = (s+3)(s+3+3i)(s+33i) ). Then, express the given fraction as the sum of two or three partial fractions. The general form for partial fraction decomposition is:
[ \frac{A}{s+3} + \frac{Bs + C}{s^2+6s+18} ]
Solve for the constants ( A ), ( B ),To integrate ( \frac{s(s+6)}{(s+3)(s^2+6s+18)} ) using partial fractions, follow these steps:
 Factor the denominator ( (s+3)(s^2+6s+18) ).
 Express the fraction as a sum of partial fractions.
 Solve for the unknown coefficients.
 Integrate each term separately.
 Combine the results.
Step 1: Factor the denominator ( (s+3)(s^2+6s+To integrate the given expression using partial fractions, first factorize the denominator: ( (s+3)(s^2+6s+18) = (s+3)(s+3+3i)(s+33i) ). Then, express the given fraction as the sum of two or three partial fractions. The general form for partial fraction decomposition is:
[ \frac{A}{s+3} + \frac{Bs + C}{s^2+6s+18} ]
Solve for the constants ( A ), ( B ), andTo integrate ( \frac{s(s+6)}{(s+3)(s^2+6s+18)} ) using partial fractions, follow these steps:
 Factor the denominator ( (s+3)(s^2+6s+18) ).
 Express the fraction as a sum of partial fractions.
 Solve for the unknown coefficients.
 Integrate each term separately.
 Combine the results.
Step 1: Factor the denominator ( (s+3)(s^2+6s+18To integrate the given expression using partial fractions, first factorize the denominator: ( (s+3)(s^2+6s+18) = (s+3)(s+3+3i)(s+33i) ). Then, express the given fraction as the sum of two or three partial fractions. The general form for partial fraction decomposition is:
[ \frac{A}{s+3} + \frac{Bs + C}{s^2+6s+18} ]
Solve for the constants ( A ), ( B ), and (To integrate ( \frac{s(s+6)}{(s+3)(s^2+6s+18)} ) using partial fractions, follow these steps:
 Factor the denominator ( (s+3)(s^2+6s+18) ).
 Express the fraction as a sum of partial fractions.
 Solve for the unknown coefficients.
 Integrate each term separately.
 Combine the results.
Step 1: Factor the denominator ( (s+3)(s^2+6s+18)To integrate the given expression using partial fractions, first factorize the denominator: ( (s+3)(s^2+6s+18) = (s+3)(s+3+3i)(s+33i) ). Then, express the given fraction as the sum of two or three partial fractions. The general form for partial fraction decomposition is:
[ \frac{A}{s+3} + \frac{Bs + C}{s^2+6s+18} ]
Solve for the constants ( A ), ( B ), and ( CTo integrate ( \frac{s(s+6)}{(s+3)(s^2+6s+18)} ) using partial fractions, follow these steps:
 Factor the denominator ( (s+3)(s^2+6s+18) ).
 Express the fraction as a sum of partial fractions.
 Solve for the unknown coefficients.
 Integrate each term separately.
 Combine the results.
Step 1: Factor the denominator ( (s+3)(s^2+6s+18) \To integrate the given expression using partial fractions, first factorize the denominator: ( (s+3)(s^2+6s+18) = (s+3)(s+3+3i)(s+33i) ). Then, express the given fraction as the sum of two or three partial fractions. The general form for partial fraction decomposition is:
[ \frac{A}{s+3} + \frac{Bs + C}{s^2+6s+18} ]
Solve for the constants ( A ), ( B ), and ( C \To integrate ( \frac{s(s+6)}{(s+3)(s^2+6s+18)} ) using partial fractions, follow these steps:
 Factor the denominator ( (s+3)(s^2+6s+18) ).
 Express the fraction as a sum of partial fractions.
 Solve for the unknown coefficients.
 Integrate each term separately.
 Combine the results.
Step 1: Factor the denominator ( (s+3)(s^2+6s+18) ):
To integrate the given expression using partial fractions, first factorize the denominator: ( (s+3)(s^2+6s+18) = (s+3)(s+3+3i)(s+33i) ). Then, express the given fraction as the sum of two or three partial fractions. The general form for partial fraction decomposition is:
[ \frac{A}{s+3} + \frac{Bs + C}{s^2+6s+18} ]
Solve for the constants ( A ), ( B ), and ( C )To integrate ( \frac{s(s+6)}{(s+3)(s^2+6s+18)} ) using partial fractions, follow these steps:
 Factor the denominator ( (s+3)(s^2+6s+18) ).
 Express the fraction as a sum of partial fractions.
 Solve for the unknown coefficients.
 Integrate each term separately.
 Combine the results.
Step 1: Factor the denominator ( (s+3)(s^2+6s+18) ):
(To integrate the given expression using partial fractions, first factorize the denominator: ( (s+3)(s^2+6s+18) = (s+3)(s+3+3i)(s+33i) ). Then, express the given fraction as the sum of two or three partial fractions. The general form for partial fraction decomposition is:
[ \frac{A}{s+3} + \frac{Bs + C}{s^2+6s+18} ]
Solve for the constants ( A ), ( B ), and ( C ) byTo integrate ( \frac{s(s+6)}{(s+3)(s^2+6s+18)} ) using partial fractions, follow these steps:
 Factor the denominator ( (s+3)(s^2+6s+18) ).
 Express the fraction as a sum of partial fractions.
 Solve for the unknown coefficients.
 Integrate each term separately.
 Combine the results.
Step 1: Factor the denominator ( (s+3)(s^2+6s+18) ):
( sTo integrate the given expression using partial fractions, first factorize the denominator: ( (s+3)(s^2+6s+18) = (s+3)(s+3+3i)(s+33i) ). Then, express the given fraction as the sum of two or three partial fractions. The general form for partial fraction decomposition is:
[ \frac{A}{s+3} + \frac{Bs + C}{s^2+6s+18} ]
Solve for the constants ( A ), ( B ), and ( C ) by equTo integrate ( \frac{s(s+6)}{(s+3)(s^2+6s+18)} ) using partial fractions, follow these steps:
 Factor the denominator ( (s+3)(s^2+6s+18) ).
 Express the fraction as a sum of partial fractions.
 Solve for the unknown coefficients.
 Integrate each term separately.
 Combine the results.
Step 1: Factor the denominator ( (s+3)(s^2+6s+18) ):
( s^To integrate the given expression using partial fractions, first factorize the denominator: ( (s+3)(s^2+6s+18) = (s+3)(s+3+3i)(s+33i) ). Then, express the given fraction as the sum of two or three partial fractions. The general form for partial fraction decomposition is:
[ \frac{A}{s+3} + \frac{Bs + C}{s^2+6s+18} ]
Solve for the constants ( A ), ( B ), and ( C ) by equating coefficientsTo integrate ( \frac{s(s+6)}{(s+3)(s^2+6s+18)} ) using partial fractions, follow these steps:
 Factor the denominator ( (s+3)(s^2+6s+18) ).
 Express the fraction as a sum of partial fractions.
 Solve for the unknown coefficients.
 Integrate each term separately.
 Combine the results.
Step 1: Factor the denominator ( (s+3)(s^2+6s+18) ):
( s^2 +To integrate the given expression using partial fractions, first factorize the denominator: ( (s+3)(s^2+6s+18) = (s+3)(s+3+3i)(s+33i) ). Then, express the given fraction as the sum of two or three partial fractions. The general form for partial fraction decomposition is:
[ \frac{A}{s+3} + \frac{Bs + C}{s^2+6s+18} ]
Solve for the constants ( A ), ( B ), and ( C ) by equating coefficients onTo integrate ( \frac{s(s+6)}{(s+3)(s^2+6s+18)} ) using partial fractions, follow these steps:
 Factor the denominator ( (s+3)(s^2+6s+18) ).
 Express the fraction as a sum of partial fractions.
 Solve for the unknown coefficients.
 Integrate each term separately.
 Combine the results.
Step 1: Factor the denominator ( (s+3)(s^2+6s+18) ):
( s^2 + To integrate the given expression using partial fractions, first factorize the denominator: ( (s+3)(s^2+6s+18) = (s+3)(s+3+3i)(s+33i) ). Then, express the given fraction as the sum of two or three partial fractions. The general form for partial fraction decomposition is:
[ \frac{A}{s+3} + \frac{Bs + C}{s^2+6s+18} ]
Solve for the constants ( A ), ( B ), and ( C ) by equating coefficients on bothTo integrate ( \frac{s(s+6)}{(s+3)(s^2+6s+18)} ) using partial fractions, follow these steps:
 Factor the denominator ( (s+3)(s^2+6s+18) ).
 Express the fraction as a sum of partial fractions.
 Solve for the unknown coefficients.
 Integrate each term separately.
 Combine the results.
Step 1: Factor the denominator ( (s+3)(s^2+6s+18) ):
( s^2 + 6To integrate the given expression using partial fractions, first factorize the denominator: ( (s+3)(s^2+6s+18) = (s+3)(s+3+3i)(s+33i) ). Then, express the given fraction as the sum of two or three partial fractions. The general form for partial fraction decomposition is:
[ \frac{A}{s+3} + \frac{Bs + C}{s^2+6s+18} ]
Solve for the constants ( A ), ( B ), and ( C ) by equating coefficients on both sides ofTo integrate ( \frac{s(s+6)}{(s+3)(s^2+6s+18)} ) using partial fractions, follow these steps:
 Factor the denominator ( (s+3)(s^2+6s+18) ).
 Express the fraction as a sum of partial fractions.
 Solve for the unknown coefficients.
 Integrate each term separately.
 Combine the results.
Step 1: Factor the denominator ( (s+3)(s^2+6s+18) ):
( s^2 + 6sTo integrate the given expression using partial fractions, first factorize the denominator: ( (s+3)(s^2+6s+18) = (s+3)(s+3+3i)(s+33i) ). Then, express the given fraction as the sum of two or three partial fractions. The general form for partial fraction decomposition is:
[ \frac{A}{s+3} + \frac{Bs + C}{s^2+6s+18} ]
Solve for the constants ( A ), ( B ), and ( C ) by equating coefficients on both sides of theTo integrate ( \frac{s(s+6)}{(s+3)(s^2+6s+18)} ) using partial fractions, follow these steps:
 Factor the denominator ( (s+3)(s^2+6s+18) ).
 Express the fraction as a sum of partial fractions.
 Solve for the unknown coefficients.
 Integrate each term separately.
 Combine the results.
Step 1: Factor the denominator ( (s+3)(s^2+6s+18) ):
( s^2 + 6s + 18 )To integrate the given expression using partial fractions, first factorize the denominator: ( (s+3)(s^2+6s+18) = (s+3)(s+3+3i)(s+33i) ). Then, express the given fraction as the sum of two or three partial fractions. The general form for partial fraction decomposition is:
[ \frac{A}{s+3} + \frac{Bs + C}{s^2+6s+18} ]
Solve for the constants ( A ), ( B ), and ( C ) by equating coefficients on both sides of the equation and then integrate each term individually.To integrate ( \frac{s(s+6)}{(s+3)(s^2+6s+18)} ) using partial fractions, follow these steps:
 Factor the denominator ( (s+3)(s^2+6s+18) ).
 Express the fraction as a sum of partial fractions.
 Solve for the unknown coefficients.
 Integrate each term separately.
 Combine the results.
Step 1: Factor the denominator ( (s+3)(s^2+6s+18) ):
( s^2 + 6s + 18 ) cannot be factored further over the real numbers.
Step 2: Express the fraction as a sum of partial fractions:
( \frac{s(s+6)}{(s+3)(s^2+6s+18)} = \frac{A}{s+3} + \frac{Bs + C}{s^2+6s+18} )
Step 3: Solve for the unknown coefficients:
( s(s+6) = A(s^2 + 6s + 18) + (Bs + C)(s+3) )
Expand and equate coefficients to find ( A ), ( B ), and ( C ).
Step 4: Integrate each term separately:
( \int \frac{A}{s+3} , ds = A \lns+3 + C_1 )
( \int \frac{Bs + C}{s^2+6s+18} , ds = \int \frac{B(s+3)}{s^2+6s+18} , ds + \int \frac{CB}{s^2+6s+18} , ds )
Use trigonometric substitution for ( \int \frac{B(s+3)}{s^2+6s+18} , ds ) and ( \int \frac{CB}{s^2+6s+18} , ds ).
Step 5: Combine the results:
Combine the integrals obtained in step 4 to get the final result.
This process will give you the integral of ( \frac{s(s+6)}{(s+3)(s^2+6s+18)} ) using partial fractions.
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When evaluating a onesided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a onesided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a onesided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a onesided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
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