int_0^1 sin left(2 tan ^{-1} sqrt{frac{1+x}{1 | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(B) Let $I = \int_0^1 \sin \left(2 	an ^{-1} \sqrt{\frac{1+x}{1-x}}ight) d x$. Substitute $x = \cos 	heta$,then $d x = -\sin 	heta d 	heta$. Whe

Vedclass · Accepted Answer

$\frac{\pi}{4}$

Vedclass · Answer

(B) Let $I = \int_0^1 \sin \left(2 	an ^{-1} \sqrt{\frac{1+x}{1-x}}ight) d x$. Substitute $x = \cos 	heta$,then $d x = -\sin 	heta d 	heta$. When $x = 0$,$	heta = \frac{\pi}{2}$. When $x = 1$,$	heta = 0$. $I = \int_{\pi/2}^0 \sin \left(2 	an ^{-1} \sqrt{\frac{1+\cos 	heta}{1-\cos 	heta}}ight) (-\sin 	heta) d 	heta$. Using $\sqrt{\frac{1+\cos 	heta}{1-\cos 	heta}} = \sqrt{\frac{2\cos^2(	heta/2)}{2\sin^2(	heta/2)}} = \cot(	heta/2) = 	an(\frac{\pi}{2} - \frac{	heta}{2})$. $I = \int_0^{\pi/2} \sin \left(2 	an^{-1} 	an(\frac{\pi}{2} - \frac{	heta}{2})ight) \sin 	heta d 	heta$. $I = \int_0^{\pi/2} \sin(\pi - 	heta) \sin 	heta d 	heta = \int_0^{\pi/2} \sin^2 	heta d 	heta$. Using $\sin^2 	heta = \frac{1-\cos 2	heta}{2}$,we get $I = \int_0^{\pi/2} \frac{1-\cos 2	heta}{2} d 	heta$. $I = \frac{1}{2} [	heta - \frac{\sin 2	heta}{2}]_0^{\pi/2} = \frac{1}{2} [(\frac{\pi}{2} - 0) - (0 - 0)] = \frac{\pi}{4}$.

$\int_0^1 \sin \left(2 \tan ^{-1} \sqrt{\frac{1+x}{1-x}}\right) d x$ is equal to :

Similar Questions

When a big drop of water is formed from $n$ small drops of water,the energy loss is $3E$,where $E$ is the energy of the bigger drop. If $R$ is the radius of the bigger drop and $r$ is the radius of the smaller drop,then the number of smaller drops $(n)$ is:

$A$ solid cylinder of density $\rho_0$, cross-section area $A$ and length $l$ floats in a liquid of density $\rho$ $(> \rho_0)$ with its axis vertical, as shown. If it is slightly displaced downward and released, the time period of oscillation will be:

The force required to move a body up a rough inclined plane is double the force required to prevent the body from sliding down the plane. The coefficient of friction,when the angle of inclination of the plane is $60^{\circ}$,is

Reproducing new plants by cells instead of seeds is known as

Which one of the following is a true peroxide?

Vedclass Test Series

Exam Paper Generator

Online Exam Module