Question

If $f\left(x\right)=\int \frac{x^2+si{n}^{2}x}{1+x^2}.se{c}^{2}xdx$ and $f\left(0\right)=0$ then $f\left(1\right)=$

• A. $1-\frac{\pi }{4}$
• B. $\frac{\pi }{4}-1$
• C. $tan1-\frac{\pi }{4}$
• D. None of these

Answer 1

Answer: (C)

$tan1-\frac{\pi }{4}$

$f\left(x\right)=\int \frac{x^2+{\sin }^{2}x}{1+x^2}.{\sec }^{2}xdx$

$f\left(x\right)=\int \frac{x^2+1-{\cos }^{2}x}{1+x^2}.{\sec }^{2}xdx$

$f\left(x\right)=\int (1-\frac{{\cos }^{2}x}{1+x^2}).{\sec }^{2}xdx$

$f\left(x\right)=\int {\sec }^{2}xdx-\int \frac{dx}{1+x^2}$

$f\left(x\right)=\tan x-{\tan }^{-1}x+C$

$\left(givenf\right(0)=0)$

$SoC=0f\left(1\right)=\tan 1-\frac{\pi }{4}$