MCQ
$\int_0^1 {{{\sin }^{ - 1}}\left( {\frac{{2x}}{{1 + {x^2}}}} \right)\,dx = } $
- ✓$\frac{\pi }{2} - 2\log \sqrt 2 $
- B$\frac{\pi }{2} + 2\log \sqrt 2 $
- C$\frac{\pi }{4} - \log \sqrt 2 $
- D$\frac{\pi }{4} + \log \sqrt 2 $
$\therefore $ $dx = {\sec ^2}\theta \,d\theta $
As $x = 1 \Rightarrow \theta = \frac{\pi }{4}$ and
$x = 0 \Rightarrow \theta = 0$, then
$I = 2\int_0^{\pi /4} {\theta {{\sec }^2}\theta \,d\theta = 2[\theta \tan \theta ]_0^{\pi /4} - 2\int_0^{\pi /4} {\tan \theta \,d\theta } } $
$= \frac{\pi }{2} + 2\,[\log \cos x]_0^{\pi /4} = \frac{\pi }{2} - 2\log \sqrt 2 $.
Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.
$g(\theta)=\sqrt{f(\theta)-1}+\sqrt{f\left(\frac{\pi}{2}-\theta\right)-1}$
where
$f(\theta)=\frac{1}{2}\left|\begin{array}{ccc}1 & \sin \theta & 1 \\ -\sin \theta & 1 & \sin \theta \\ -1 & -\sin \theta & 1\end{array}\right|+\left|\begin{array}{ccc}\sin \pi & \cos \left(\theta+\frac{\pi}{4}\right) & \tan \left(\theta-\frac{\pi}{4}\right) \\ \sin \left(\theta-\frac{\pi}{4}\right) & -\cos \frac{\pi}{2} & \log _e\left(\frac{4}{\pi}\right) \\ \cot \left(\theta+\frac{\pi}{4}\right) & \log _e\left(\frac{\pi}{4}\right) & \tan \pi\end{array}\right|$.
Let $p (x)$ be a quadratic polynomial whose roots are the maximum and minimum values of the function $g(\theta)$, and $p(2)=2-\sqrt{2}$. Then, which of the following is/are TRUE ?
$(A)$ $p \left(\frac{3+\sqrt{2}}{4}\right)<0$
$(B)$ $p \left(\frac{1+3 \sqrt{2}}{4}\right)>0$
$(C)$ $p \left(\frac{5 \sqrt{2}-1}{4}\right)>0$
$(D)$ $p \left(\frac{5-\sqrt{2}}{4}\right)<0$