The integral $\int \frac { x } { x ^ { 2 } + 2 x + 10 } d x$ is

A) $- \frac { 3 \ln \left| x ^ { 2 } + 2 x + 10 \right| + 2 \tan ^ { - 1 } \left( \frac { x + 1 } { 3 } \right) } { 6 } + C$
B) $\frac { 3 \ln \left| x ^ { 2 } + 2 x + 10 \right| - 2 \tan ^ { - 1 } \left( \frac { x + 1 } { 3 } \right) } { 6 } + C$
C) $- \frac { 3 \ln \left| x ^ { 2 } + 2 x + 10 \right| - 2 \tan ^ { - 1 } \left( \frac { x + 1 } { 3 } \right) } { 3 } + C$
D) $\frac { 3 \ln \left| x ^ { 2 } + 2 x + 10 \right| - 2 \tan ^ { - 1 } \left( \frac { x + 1 } { 3 } \right) } { 3 } + C$
E) $\frac { 3 \ln \left| x ^ { 2 } + 2 x + 10 \right| - 2 \tan ^ { - 1 } \left( \frac { x + 1 } { 3 } \right) } { 12 } + C$

Question

The integral $\int \frac { x } { x ^ { 2 } + 2 x + 10 } d x$ is

A) $- \frac { 3 \ln \left| x ^ { 2 } + 2 x + 10 \right| + 2 \tan ^ { - 1 } \left( \frac { x + 1 } { 3 } \right) } { 6 } + C$
B) $\frac { 3 \ln \left| x ^ { 2 } + 2 x + 10 \right| - 2 \tan ^ { - 1 } \left( \frac { x + 1 } { 3 } \right) } { 6 } + C$
C) $- \frac { 3 \ln \left| x ^ { 2 } + 2 x + 10 \right| - 2 \tan ^ { - 1 } \left( \frac { x + 1 } { 3 } \right) } { 3 } + C$
D) $\frac { 3 \ln \left| x ^ { 2 } + 2 x + 10 \right| - 2 \tan ^ { - 1 } \left( \frac { x + 1 } { 3 } \right) } { 3 } + C$
E) $\frac { 3 \ln \left| x ^ { 2 } + 2 x + 10 \right| - 2 \tan ^ { - 1 } \left( \frac { x + 1 } { 3 } \right) } { 12 } + C$

Quizplus · Accepted Answer

The integral can be solved by completing the square in the denominator and then using partial fraction decomposition or recognizing the derivative of arctan function. Completing the square for $x^2 + 2x + 10$ gives $(x+1)^2 + 9$, which suggests a substitution related to the arctan function. The correct answer involves the logarithm of the quadratic expression and the arctan of a linear expression, with coefficients adjusted to match the derivative of the completed square form.

The Integral ∫xx2+2x+10dx\int \frac { x } { x ^ { 2 } + 2 x + 10 } d x∫x2+2x+10x​dx

The Integral $\int \frac { x } { x ^ { 2 } + 2 x + 10 } d x$