Question

$\int \frac{{(x+1)}^2}{x(x^2+1)}dx$ is equal to

• A. $\log \left|x(x^2+1)\right|+C$
• B. $\log \left|x\right|+C$
• C. $\log \left|x\right|+2{\tan }^{-1}x+C$
• D. $\log \left(\frac{1}{1+x^2}\right)+C$
• E. $2\log \left|x\right|+{\tan }^{-1}x+C$

Answer 1

Answer: (C)

$\log \left|x\right|+2{\tan }^{-1}x+C$

Let $I=\int \frac{{(x+1)}^2}{x(x^2+1)}dx$

$=\int \frac{x^2+1+2x}{x(x^2+1)}dx$

$=\int \frac{x^2+1}{x(x^2+1)}dx+\int \frac{2x}{x(x^2+1)}dx$

$=\int \frac{1}{x}dx+2\int \frac{1}{x^2+1}dx$

$=\log \left|x\right|+2{\tan }^{-1}x+C$