Practicing Success
If $f(x)=\int\frac{(x^2+\sin^2x)}{1+x^2}\sec^2x\,dx$ and f(0) = 0, then f(1) is: |
$1-\frac{π}{4}$ $\frac{π}{4}-1$ $\tan 1-\frac{π}{4}$ None of these |
$\tan 1-\frac{π}{4}$ |
We have $f(x)=\int\frac{(x^2+\sin^2x)}{1+x^2}\sec^2x\,dx$ $=\int\frac{x^2+(1-\cos^2x)}{1+x^2}.\sec^2x\,dx=\int(\sec^2x-\frac{1}{1+x^2})dx$ Thus, $f(x)=\tan x-\tan^{-1}x+C$ As $f(0)=0,C=0⇒f(x)=\tan x-\tan^{-1}x$ Hence, $f(1)=\tan 1-tan^{-1}1=\tan 1-\frac{π}{4}$ |