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

Distributing and simplifying, we get
\begin{align*}
\left(\frac{5-x}{x}\right)^2&=\frac{(5-x)^2}{x^2}\
&=\frac{25-10x+x^2}{x^2}\
&=\frac{25}{x^2}-\frac{10}{x}+1
\end{align*}
Integrating each term separately, we get
$$\int\frac{25}{x^2}dx=-\frac{25}{x},\quad \int\frac{10}{x}dx=10\ln|x|,\quad \int dx=x$$
Therefore,
$$\int\left(\frac{5-x}{x}\right)^2dx=\int\left(\frac{25}{x^2}-\frac{10}{x}+1\right)dx=-\frac{25}{x}+10\ln|x|+x+C$$
which matches answer choice $\boxed{\textbf{(B)}}$.

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