Express the integrand as a sum of partial fractions and evaluate the integral.
-$$\int \frac { x ^ { 3 } } { x ^ { 2 } + 4 x + 4 } d x$$
A) $\frac { x ^ { 2 } } { 2 } - 4 x + 12 \ln | x + 2 | + \frac { 8 } { x + 2 } + C$
B) $\frac { x ^ { 2 } } { 2 } - 4 x + 6 \ln | x + 2 | - \frac { 4 } { x + 2 } + C$
C) $12 \ln | x - 4 | + \frac { 12 } { x + 2 } - \frac { 8 } { ( x + 2 ) ^ { 2 } } + C$
D) $\frac { x ^ { 2 } } { 2 } - 4 x - 12 \ln | x + 2 | + \frac { 8 } { ( x + 2 ) ^ { 2 } } + C$

Question

Quizplus · Accepted Answer

The integrand $\frac{x^3}{x^2 + 4x + 4}$ can be simplified by dividing $x^3$ by $x^2 + 4x + 4$, which results in $x - 4 + \frac{16}{x^2 + 4x + 4}$. The denominator can be factored as $(x+2)^2$, leading to the integral of $x - 4 + \frac{16}{(x+2)^2}$. This can be integrated directly to get $\frac{x^2}{2} - 4x + 16 \cdot \frac{1}{x+2}$ plus the integration constant $C$, which matches the form of option A after simplifying.

Express the Integrand as a Sum of Partial Fractions and Evaluate