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

A) $\frac { 2 \ln | x + 2 | + 4 \tan ^ { - 1 } ( x ) - \ln \left| x ^ { 2 } + 1 \right| } { 10 } + C$
B) $\frac { 2 \ln | x + 2 | - 4 \tan ^ { - 1 } ( x ) - \ln \left| x ^ { 2 } + 1 \right| } { 10 } + C$
C) $\frac { 2 \ln | x + 2 | + 4 \tan ^ { - 1 } ( x ) + \ln \left| x ^ { 2 } + 1 \right| } { 10 } + C$
D) $- \frac { 2 \ln | x + 2 | + 4 \tan ^ { - 1 } ( x ) - \ln \left| x ^ { 2 } + 1 \right| } { 10 } + C$
E) $- \frac { 2 \ln | x + 2 | - 4 \tan ^ { - 1 } ( x ) + \ln \left| x ^ { 2 } + 1 \right| } { 10 } + C$

Question

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

A) $\frac { 2 \ln | x + 2 | + 4 \tan ^ { - 1 } ( x ) - \ln \left| x ^ { 2 } + 1 \right| } { 10 } + C$
B) $\frac { 2 \ln | x + 2 | - 4 \tan ^ { - 1 } ( x ) - \ln \left| x ^ { 2 } + 1 \right| } { 10 } + C$
C) $\frac { 2 \ln | x + 2 | + 4 \tan ^ { - 1 } ( x ) + \ln \left| x ^ { 2 } + 1 \right| } { 10 } + C$
D) $- \frac { 2 \ln | x + 2 | + 4 \tan ^ { - 1 } ( x ) - \ln \left| x ^ { 2 } + 1 \right| } { 10 } + C$
E) $- \frac { 2 \ln | x + 2 | - 4 \tan ^ { - 1 } ( x ) + \ln \left| x ^ { 2 } + 1 \right| } { 10 } + C$

Quizplus · Accepted Answer

To solve the integral $\int \frac { 1 } { ( x + 2 ) \left( x ^ { 2 } + 1 \right) } d x$, partial fraction decomposition is used, leading to terms that integrate to $\ln |x+2|$, $\tan^{-1}(x)$, and $\ln |x^2+1|$. The coefficients and signs in option A correctly reflect the results of integrating these terms.

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

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