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

We can rewrite the integrand as
$$\left(\frac{5}{x}-1\right)^2 = \frac{25}{x^2}-\frac{10}{x}+1.$$
Then we can integrate term by term to get
\begin{align*}
\int \left( \frac { 5 - x } { x } \right) ^ { 2 } d x &= \int \left(\frac{25}{x^2}-\frac{10}{x}+1\right)dx \
&= -\frac{25}{x} - 10\ln|x| + x + C \
&= x - 10\ln|x| - \frac{25}{x} + C.
\end{align*}
Therefore, the answer is choice $\boxed{\text{(B)}}$.

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 $\int \left( \frac { 5 - x } { x } \right) ^ { 2 } d x$