Evaluate the integral.
$$\int 4 \left( \frac { x - 1 } { x ^ { 2 } + 2 x } \right) d x$$
A) $4 \left( \frac { 1 } { 3 } x - \frac { 1 } { 3 } \ln x \right) + C$
B) $4 \left( \frac { 3 } { 2 } \ln ( x + 2 ) - \frac { 1 } { 2 } \ln x \right) + C$
C) $- 4 \left( \frac { \sqrt { 2 } } { 2 } \arctan x \right) + C$
D) $- 4 ( \ln x - x ) + C$
E) None of these

Question

Quizplus · Accepted Answer

Using the substitution $u = x^2 + 2x$, we get $du = (2x + 2) dx = 2(x + 1) dx$. So,$$\int 4 \left( \frac { x - 1 } { x ^ { 2 } + 2 x } \right) d x = \int 4 \left( \frac { x - 1 } { u } \right) \frac { du } { 2 ( x + 1 ) } = 2 \int \frac { 2 x - 2 } { u } d u = 2 \int \left( \frac { 2 } { u } - \frac { 2 x } { u } \right) d u = 2 \left( 2 \ln | u | - \ln | x + 2 | \right) + C = 4 \left( \ln | x ^ { 2 } + 2 x | - \frac { 1 } { 2 } \ln | x + 2 | \right) + C = 4 \left( \frac { 3 } { 2 } \ln ( x + 2 ) - \frac { 1 } { 2 } \ln x \right) + C$$

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

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