Question

$\int\left(\frac{x-a}{x}-\frac{x}{x+a}\right) dx$ is equal to

• A. $\log\left|\frac{x+a}{x}\right|+C$
• B. a$\log\left|\frac{x+a}{x}\right|+C$
• C. a$\log\left|\frac{x}{x+a}\right|+C$
• D. $\log\left|\frac{x}{x+a}\right|+C$
• E. $a\log\left|\frac{x-a}{x+a}\right|+C$

Answer 1

Answer: (B)

Let $ I=\int\left(\frac{x-a}{x}-\frac{x}{x+a}\right) d x $
$=\int \frac{\left(x^{2}-a^{2}\right)-x^{2}}{x(x+a)} d x $
$=-a^{2} \int \frac{1}{x(x+a)} d x $
$=\frac{-a^{2}}{a} \int\left[\frac{1}{x}-\frac{1}{x+a}\right] d x$
$=-a \log \left|\frac{x}{x+a}\right|+C=a \log \left|\frac{x+a}{x}\right|+C$