Express the integrand as a sum of partial fractions and evaluate the integral.
-$$\int \frac { 4 x ^ { 3 } + 20 x ^ { 2 } + 32 x + 13 } { x ^ { 2 } \left( x ^ { 2 } + 6 x + 13 \right) } d x$$
A) $7 \ln | x | + \ln \left| x ^ { 2 } + 6 x + 13 \right| + \frac { 1 } { 2 } \tan ^ { - 1 } \left( \frac { x + 3 } { 2 } \right) + C$
B) $2 \ln | x | - \frac { 1 } { x } + \ln \left| x ^ { 2 } + 6 x + 13 \right| + \frac { 1 } { 2 } \tan ^ { - 1 } \left( \frac { x + 3 } { 2 } \right) + C$
C) $2 \ln \left| x - \frac { 1 } { x } \right| + \ln \left| x ^ { 2 } + 6 x + 13 \right| + \frac { 1 } { 2 } \tan ^ { - 1 } \left( \frac { x + 3 } { 2 } \right) + C$
D) $5 \ln | x | + \frac { 1 } { x } - \ln \left| x ^ { 2 } + 6 x + 13 \right| + \sin ^ { - 1 } \left( \frac { x + 3 } { 2 } \right) + C$

Question

Quizplus · Accepted Answer

The integrand can be decomposed into partial fractions, and upon integration, the terms correspond to the logarithmic and inverse trigonometric functions as shown in choice B.

Express the Integrand as a Sum of Partial Fractions and Evaluate