The integral $\int \frac { x - 1 } { x ^ { 2 } + 2 x + 10 } d x$ is

A) $- \frac { 3 \ln \left| x ^ { 2 } + 2 x + 10 \right| - 4 \tan ^ { - 1 } \left( \frac { x + 1 } { 3 } \right) } { 3 } + C$
B) $\frac { 3 \ln \left| x ^ { 2 } + 2 x + 10 \right| + 4 \tan ^ { - 1 } \left( \frac { x + 1 } { 3 } \right) } { 6 } + C$
C) $\frac { 3 \ln \left| x ^ { 2 } + 2 x + 10 \right| - 4 \tan ^ { - 1 } \left( \frac { x + 1 } { 3 } \right) } { 3 } + C$
D) $\frac { 3 \ln \left| x ^ { 2 } + 2 x + 10 \right| - 4 \tan ^ { - 1 } \left( \frac { x + 1 } { 3 } \right) } { 6 } + C$
E) $\frac { 3 \ln \left| x ^ { 2 } + 2 x + 10 \right| + 4 \tan ^ { - 1 } \left( \frac { x + 1 } { 3 } \right) } { 12 } + C$

Question

The integral $\int \frac { x - 1 } { x ^ { 2 } + 2 x + 10 } d x$ is

A) $- \frac { 3 \ln \left| x ^ { 2 } + 2 x + 10 \right| - 4 \tan ^ { - 1 } \left( \frac { x + 1 } { 3 } \right) } { 3 } + C$
B) $\frac { 3 \ln \left| x ^ { 2 } + 2 x + 10 \right| + 4 \tan ^ { - 1 } \left( \frac { x + 1 } { 3 } \right) } { 6 } + C$
C) $\frac { 3 \ln \left| x ^ { 2 } + 2 x + 10 \right| - 4 \tan ^ { - 1 } \left( \frac { x + 1 } { 3 } \right) } { 3 } + C$
D) $\frac { 3 \ln \left| x ^ { 2 } + 2 x + 10 \right| - 4 \tan ^ { - 1 } \left( \frac { x + 1 } { 3 } \right) } { 6 } + C$
E) $\frac { 3 \ln \left| x ^ { 2 } + 2 x + 10 \right| + 4 \tan ^ { - 1 } \left( \frac { x + 1 } { 3 } \right) } { 12 } + C$

Quizplus · Accepted Answer

The integral can be solved by splitting the fraction into two parts, one that directly integrates to a logarithm and another that integrates to an arctangent function. Completing the square in the denominator gives $x^2 + 2x + 10 = (x+1)^2 + 9$, which helps in identifying the arctangent part. The correct answer involves both a logarithmic and an arctangent term, with coefficients adjusted to match the derivative of the denominator and the form required for the arctangent integral. The division by 6 is due to the coefficients involved in the integration process.

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

The Integral $\int \frac { x - 1 } { x ^ { 2 } + 2 x + 10 } d x$