Evaluate the integral if it exists. $$\int \left( \frac { 5 - x } { x } \right) ^ { 2 } d x$$
A) $1 - 10 \ln x + C$
B)
$x - 10 \ln x - \frac { 25 } { x } + C$
C) $\ln x - 25 x + C$
D)
$x - \frac { 1 } { 10 \ln x } + C$
E) none of these

Question

Quizplus · Accepted Answer

To solve the integral, we can use u-substitution, where $u=5-x$. Then, $du=-dx$ and the integral becomes:

\begin{align*}
\int \left(\frac{5-x}{x}ight)^2 dx &= -\int \left(\frac{u}{5-u}ight)^2 du \
&= -\int\frac{u^2}{(5-u)^2}du \
&= -\int\left(\frac{25}{(5-u)^2}-\frac{10}{5-u}+1ight)du \
&= -\left(-\frac{25}{5-u}+10\ln|5-u|-uight)+C \
&= x-10\ln|x-5|-\frac{25}{x}+C \
\end{align*}

The correct choice is (B).

Evaluate the Integral If It Exists ∫(5−xx)2dx\int \left( \frac { 5 - x } { x } \right) ^ { 2 } d x∫(x5−x​)2dx

Evaluate the Integral If It Exists $\int \left( \frac { 5 - x } { x } \right) ^ { 2 } d x$