Express the integrand as a sum of partial fractions and evaluate the integral.
-$$\int \frac { 7 x ^ { 3 } + 35 x ^ { 2 } + 55 x + 23 } { ( x + 3 ) ( x + 1 ) ^ { 3 } } d x$$
A) $\ln \left| ( x + 3 ) ^ { 2 } ( x + 1 ) ^ { 5 } \right| + \frac { 3 } { ( x + 1 ) ^ { 2 } } + C$
B) $\ln \left| ( x + 3 ) ^ { 2 } ( x + 1 ) ^ { 5 } \right| - \frac { 4 } { ( x + 1 ) } + \frac { 1 } { ( x + 1 ) ^ { 2 } } + C$
C) $\ln \left| ( x + 3 ) ^ { 2 } ( x + 1 ) ^ { 5 } \right| - \frac { 1 } { ( x + 1 ) } + \frac { 4 } { ( x + 1 ) ^ { 2 } } + C$
D) $\ln \left| ( x + 3 ) ^ { 2 } ( x + 1 ) ^ { 5 } \right| - \frac { 5 } { ( x + 1 ) ^ { 2 } } + C$

Question

Quizplus · Accepted Answer

The correct partial fraction decomposition leads to terms that include a natural logarithm function for the polynomial part and individual terms for the repeated linear factors. The given integral, when decomposed, will not result in a simple power expression as in choices A, C, and D, but rather a combination of logarithmic and rational functions, reflecting the structure of the denominator.

Express the Integrand as a Sum of Partial Fractions and Evaluate