Express the Integrand as a Sum of Partial Fractions and Evaluate

question 209

Multiple Choice

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) \(5 \ln | x | + \frac { 1 } { x } - \ln \left| x ^ { 2 } + 6 x + 13 \right| + \sin ^ { - 1 } \left( \frac { x + 3 } { 2 } \right) + C\)