Question

$\int e^x\left(\frac{2}{x}-\frac{2}{x^2}\right)dx$ is equal to

• A. $\frac{e^x}{x}+C$
• B. $\frac{e^x}{2x^2}+C$
• C. $\frac{2e^x}{x}+C$
• D. $\frac{2e^x}{x^2}+C$

Answer 1

Answer: (C)

$I=\int e^x\left(\frac{2}{x}-\frac{2}{x^2}\right)dx$
$=2\int e^x\frac{1}{x}dx-2\int \frac{e^x}{x^2}dx$
$=2\cdot \left[\frac{1}{x}\cdot e^x-\int -\frac{1}{x^2}{e}^{x}dx\right]-2\int \frac{e^x}{x^2}dx+c$
$=\frac{2e^x}{x}+c$