# How do you integrate #(6x+5) / (x+2) ^4# using partial fractions?

I feel that there is no need to use the Method of Partial Fractions,

because, without it, the Problem can be easily worked out, as shown

below:

We know that,

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To integrate ( \frac{6x + 5}{(x + 2)^4} ) using partial fractions, follow these steps:

Step 1: Perform polynomial division (if necessary) to ensure that the degree of the numerator is less than the degree of the denominator.

Step 2: Express ( \frac{6x + 5}{(x + 2)^4} ) as a sum of partial fractions. Since the denominator ( (x + 2)^4 ) has repeated factors, the decomposition will involve terms of the form ( \frac{A}{(x + 2)^n} ), where ( n ) is the multiplicity of the factor ( (x + 2) ).

Step 3: Solve for the unknown constants ( A ) using algebraic methods such as equating coefficients or substitution.

Step 4: After finding the partial fraction decomposition, integrate each term separately.

Given that the denominator ( (x + 2)^4 ) has repeated factors, the decomposition will involve terms of the form ( \frac{A}{(x + 2)^n} ), where ( n = 1, 2, 3, 4 ).

Let's denote:

[ \frac{6x + 5}{(x + 2)^4} = \frac{A}{x + 2} + \frac{B}{(x + 2)^2} + \frac{C}{(x + 2)^3} + \frac{D}{(x + 2)^4} ]

Now, find the values of ( A ), ( B ), ( C ), and ( D ).

After finding the values of ( A ), ( B ), ( C ), and ( D ), integrate each term separately. This will yield the integral of the original function.

Please note that the process of solving for the unknown constants ( A ), ( B ), ( C ), and ( D ) involves algebraic manipulation, and the values will depend on the specific coefficients of the original function.

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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.

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