The integral $\int \frac { 7 x + 3 } { x ^ { 3 } - 2 x ^ { 2 } - 3 x } d x$ is

A) $\ln \left| \frac { ( x - 3 ) ^ { 2 } } { x ( x + 1 ) } \right| + C$
B) $- \ln \left| \frac { ( x - 3 ) ^ { 2 } } { x ( x + 1 ) } \right| + C$
C) $2 \ln \left| \frac { ( x - 3 ) ^ { 2 } } { x ( x + 1 ) } \right| + C$
D) $- 2 \ln \left| \frac { ( x - 3 ) ^ { 2 } } { x ( x + 1 ) } \right| + C$
E) $3 \ln \left| \frac { ( x - 3 ) ^ { 2 } } { x ( x + 1 ) } \right| + C$

Question

The integral $\int \frac { 7 x + 3 } { x ^ { 3 } - 2 x ^ { 2 } - 3 x } d x$ is

A) $\ln \left| \frac { ( x - 3 ) ^ { 2 } } { x ( x + 1 ) } \right| + C$
B) $- \ln \left| \frac { ( x - 3 ) ^ { 2 } } { x ( x + 1 ) } \right| + C$
C) $2 \ln \left| \frac { ( x - 3 ) ^ { 2 } } { x ( x + 1 ) } \right| + C$
D) $- 2 \ln \left| \frac { ( x - 3 ) ^ { 2 } } { x ( x + 1 ) } \right| + C$
E) $3 \ln \left| \frac { ( x - 3 ) ^ { 2 } } { x ( x + 1 ) } \right| + C$

Quizplus · Accepted Answer

The integral can be solved by partial fraction decomposition. The denominator factors as $x(x-3)(x+1)$. The partial fraction decomposition leads to a solution involving logarithms, and after simplification, the correct answer is option A.

The Integral ∫7x+3x3−2x2−3xdx\int \frac { 7 x + 3 } { x ^ { 3 } - 2 x ^ { 2 } - 3 x } d x∫x3−2x2−3x7x+3​dx

The Integral $\int \frac { 7 x + 3 } { x ^ { 3 } - 2 x ^ { 2 } - 3 x } d x$