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

A) $- \frac { \ln \left| ( x + 1 ) ( x - 3 ) ^ { 3 } \right| } { 4 } + C$
B) $\frac { \ln \left| ( x + 1 ) ( x - 3 ) ^ { 3 } \right| } { 4 } + C$
C) $\frac { \ln \left| ( x + 1 ) ( x - 3 ) ^ { 3 } \right| } { 2 } + C$
D) $- \frac { \ln \left| ( x + 1 ) ( x - 3 ) ^ { 3 } \right| } { 2 } + C$
E) $\frac { \ln \left| ( x + 1 ) ( x - 3 ) ^ { 3 } \right| } { 8 } + C$

Question

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

A) $- \frac { \ln \left| ( x + 1 ) ( x - 3 ) ^ { 3 } \right| } { 4 } + C$
B) $\frac { \ln \left| ( x + 1 ) ( x - 3 ) ^ { 3 } \right| } { 4 } + C$
C) $\frac { \ln \left| ( x + 1 ) ( x - 3 ) ^ { 3 } \right| } { 2 } + C$
D) $- \frac { \ln \left| ( x + 1 ) ( x - 3 ) ^ { 3 } \right| } { 2 } + C$
E) $\frac { \ln \left| ( x + 1 ) ( x - 3 ) ^ { 3 } \right| } { 8 } + C$

Quizplus · Accepted Answer

The integral can be solved by partial fraction decomposition. The denominator $x^2 - 2x - 3$ factors into $(x - 3)(x + 1)$. Setting up and solving the partial fraction decomposition leads to an expression that, when integrated, results in the natural logarithm of the absolute value of the product of $(x + 1)$ and $(x - 3)$, with the coefficients determined by the decomposition. The correct answer involves the natural logarithm with a coefficient of $\frac{1}{4}$, which matches option B.

The Integral ∫xx2−2x−3dx\int \frac { x } { x ^ { 2 } - 2 x - 3 } d x∫x2−2x−3x​dx

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