How do you integrate #f(x)=1/((x-2)(x-5)(x+3))# using partial fractions?
now, multiply through by (x-2)(x-5)(x+3)
so 1 = A(x-5)(x+3) + B(x-2)(x+3) + C(x-2)(x-5) ................................(1)
We now have to find the values of A , B and C. Note that if x = 2 , the terms with B and C will be zero. If x = 5 , the terms with A and C will be zero and if x = -3, the terms with A and B will be zero. Making use of this fact , we obtain.
Integral now becomes.
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To integrate ( f(x) = \frac{1}{(x-2)(x-5)(x+3)} ) using partial fractions, follow these steps:
- Factor the denominator if necessary.
- Write ( f(x) ) as a sum of partial fractions.
- Determine the constants for each partial fraction.
- Integrate each partial fraction separately.
- Combine the results to find the final integral.
Here are the detailed steps:
-
Factor the denominator: ( (x-2)(x-5)(x+3) ) cannot be further factored since it's a product of distinct linear factors.
-
Write ( f(x) ) as partial fractions: ( f(x) = \frac{A}{x-2} + \frac{B}{x-5} + \frac{C}{x+3} )
-
Determine the constants ( A ), ( B ), and ( C ): To find ( A ), ( B ), and ( C ), you can use methods like equating coefficients or substitution.
-
Integrate each partial fraction separately:
- ( \int \frac{A}{x-2} , dx = A \ln|x-2| + C_1 )
- ( \int \frac{B}{x-5} , dx = B \ln|x-5| + C_2 )
- ( \int \frac{C}{x+3} , dx = C \ln|x+3| + C_3 )
-
Combine the results: ( \int f(x) , dx = A \ln|x-2| + B \ln|x-5| + C \ln|x+3| + C_1 + C_2 + C_3 )
Where ( A ), ( B ), and ( C ) are the constants determined in step 3, and ( C_1 ), ( C_2 ), and ( C_3 ) are integration constants.
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When evaluating a one-sided 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 one-sided 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 one-sided 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 one-sided 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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