How do you integrate #int(x+1)/((x^2-8)(x+4))# using partial fractions?

Question

Elijah Abram · Accepted Answer

Derived below

Samantha Adkison · Answer

undefined To integrate $ \frac{{\text{int}(x+1)}}{{(x^2 - 8)(x+4)}} $ using partial fractions, first express the given expression as the sum of two fractions with simpler denominators, namely $ (x^2 - 8) $ and $ (x+4) $. Then, find the constants to represent these fractions.

Given expression: $ \frac{{\text{int}(x+1)}}{{(x^2 - 8)(x+4)}} $

Step 1: Factorize the denominator:
$ x^2 - 8 = (x + 2)(x - 2) $

Step 2: Express the original fraction as:
$ \frac{{A}}{{x - 2}} + \frac{{B}}{{x + 2}} + \frac{{C}}{{x + 4}} $

Step 3: Multiply both sides by the original denominator to clear it:
$ \text{int}(x+1) = A(x+4)(x+2) + B(x+4)(x-2) + C(x-2)(x+2) $

Step 4: Solve for the constants A, B, and C by comparing coefficients.

Step 5: After finding the constants, integrate each fraction separately.

Step 6: Finally, sum up the integrals of each fraction to get the final result.