The integral $\int \frac { 10 } { ( 1 - x ) \left( x ^ { 2 } + 4 \right) } d x$ is

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

Question

The integral $\int \frac { 10 } { ( 1 - x ) \left( x ^ { 2 } + 4 \right) } d x$ is

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

Quizplus · Accepted Answer

The integral can be solved by partial fraction decomposition and recognizing the integral forms of natural logarithm and arctangent functions. The correct answer involves a positive logarithmic term and a positive arctangent term, which matches option B.

The Integral ∫10(1−x)(x2+4)dx\int \frac { 10 } { ( 1 - x ) \left( x ^ { 2 } + 4 \right) } d x∫(1−x)(x2+4)10​dx

The Integral $\int \frac { 10 } { ( 1 - x ) \left( x ^ { 2 } + 4 \right) } d x$