Question

Evaluate: $\int x\frac{d}{dx}\left(\ln \right(1+x^2\left)\right)dx$

• A. $2\left({\tan }^{-1}x\right)+c$
• B. $2(x-{\tan }^{-1}x)+c$
• C. $2(\ln x-{\tan }^{-1}x)+c$
• D. $2\left(\ln \right(1+x^2)-{\tan }^{-1}x)+c$

Answer 1

Answer: (B)

$2(x-{\tan }^{-1}x)+c$

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

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

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

$=2x-2{\tan }^{-1}x+c$

$=2(x-{\tan }^{-1}x)+c$

Hence, option B is correct.