Express the integrand as a sum of partial fractions and evaluate the integral.
-$$\int \frac { 2 x ^ { 3 } + 5 x ^ { 2 } + 14 x + 7 } { \left( x ^ { 2 } + 2 x + 5 \right) ^ { 2 } } d x$$
A) $\ln \left| x ^ { 2 } + 2 x + 5 \right| - \frac { 1 } { 2 } \tan ^ { - 1 } \left( \frac { x + 1 } { 2 } \right) + C$
B) $\ln \left| x ^ { 2 } + 2 x + 5 \right| - \frac { 1 } { 2 } \tan ^ { - 1 } \left( \frac { x + 1 } { 2 } \right) - \frac { 1 } { x ^ { 2 } + 2 x + 5 } + C$
C) $- \frac { 1 } { 2 } \tan ^ { - 1 } \left( \frac { \mathrm { x } + 1 } { 2 } \right) - \frac { 1 } { \mathrm { x } ^ { 2 } + 2 \mathrm { x } + 5 } + \mathrm { C }$
D) $\ln \left| x ^ { 2 } + 2 x + 5 \right| - \frac { 1 } { x ^ { 2 } + 2 x + 5 } + C$

Question

Quizplus · Accepted Answer

The integral can be solved by expressing the integrand as a sum of partial fractions, which involves breaking it down into simpler fractions that can be integrated individually. The correct expression involves a logarithmic function, an arctan function, and a term with the denominator $x^2 + 2x + 5$, which matches option B.

Express the Integrand as a Sum of Partial Fractions and Evaluate