Evaluate the integral by first performing long division on the integrand and then writing the proper fraction as a sum of partial fractions.
-$$\int \frac { x ^ { 3 } } { x ^ { 2 } + 2 x + 1 } d x$$
A) $3 \ln | x - 2 | + \frac { 3 } { x + 1 } - \frac { 1 } { ( x + 1 ) ^ { 2 } } + C$
B) $\frac { x ^ { 2 } } { 2 } - 2 x + 3 \ln | x + 1 | + \frac { 1 } { x + 1 } + C$
C) $\frac { x ^ { 2 } } { 2 } - 2 x - 3 \ln | x + 1 | + \frac { 1 } { ( x + 1 ) ^ { 2 } } + C$
D) $\frac { x ^ { 2 } } { 2 } - 2 x + 3 \ln | x + 1 | - \frac { 1 } { x + 1 } + C$

Question

Quizplus · Accepted Answer

First, perform long division on the integrand to get $x - 2 + \frac{3}{x^2 + 2x + 1}$. Then, factor the denominator as $(x+1)^2$, and use partial fractions if necessary. The integral simplifies to $\frac{x^2}{2} - 2x + 3\ln|x+1| + \frac{1}{x+1} + C$.

Evaluate the Integral by First Performing Long Division on the Integrand