Find the indefinite integral $\int \frac { x ^ { 2 } - 25 } { x ^ { 2 } + 25 } d x$.
A) $\frac { x ^ { 3 } - 25 x } { x ^ { 3 } + 25 x } + C$
B) $x - 10 \arctan \left( \frac { x } { 5 } \right) + C$
C) $x - 6 \arctan \left( \frac { x } { 5 } \right) + C$
D) $\frac { x ^ { 3 } - 75 x } { x ^ { 3 } + 75 x } + C$

Question

Quizplus · Accepted Answer

We can use the substitution $u=x^2+25$ and $du=2x\,dx$. Then,
\begin{align*} \int \frac{x^2-25}{x^2+25}\,dx &= \frac{1}{2} \int \frac{u-50}{u}\,du \ &= \frac 12\int \left(1-\frac{50}{u}ight)\,du \ &= \frac{1}{2}\left(u-50\ln\left|uight|ight) + C \ &= \frac{1}{2}\left(x^2-25-50\ln\left|x^2+25ight|ight) + C \ &= x-10\arctan\left(\frac{x}{5}ight) + C. \end{align*} Therefore, the answer is $\boxed{	extbf{(B)}}$.

Find the Indefinite Integral ∫x2−25x2+25dx\int \frac { x ^ { 2 } - 25 } { x ^ { 2 } + 25 } d x∫x2+25x2−25​dx

Find the Indefinite Integral $\int \frac { x ^ { 2 } - 25 } { x ^ { 2 } + 25 } d x$