Question

$\int (\frac{x-a}{x}-\frac{x}{x+a})dx$ 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)

$a\log \left|\frac{x+a}{x}\right|+C$

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