Question

$\int \frac{x^2+co{s}^{2}x}{(1+x^2)si{n}^{2}x}dx$ equals to

• A. $ta{n}^{-1}x+cotx+c$
• B. $ta{n}^{-1}x-cotx+c$
• C. $co{t}^{-1}x-tanx+c$
• D. $-ta{n}^{-1}x-cotx+c$

Answer 1

The correct option is D $-ta{n}^{-1}x-cotx+c$

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

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

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

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

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