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

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

Question

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

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

Quizplus · Accepted Answer

The integral can be solved by recognizing that the denominator is already a complete square, $x^2 + 2x + 10 = (x+1)^2 + 9$, and the numerator is the derivative of the denominator's quadratic term, $x+1$. This makes the integral take the form of the derivative of a natural log, leading to $\frac { \ln \left| x ^ { 2 } + 2 x + 10 \right| } { 2 } + C$.

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

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