Question
Evaluate: $\int_{\frac{1}{\sqrt{2}}}^1 \frac{\left( e ^{\left.\cos ^{-1} x\right)\left(\sin ^{-1} x\right)}\right.}{\sqrt{1-x^2}} d x$
✓
Answer
$ \text { Let } I =\int_{\frac{1}{\sqrt{2}}}^1 \frac{\left( e ^{\cos ^{-1} x}\right)\left(\sin ^{-1} x\right)}{\sqrt{1-x^2}} d x$
$=\int_{\frac{1}{\sqrt{2}}}^1 \frac{\left( e ^{\left.\frac{\pi}{2}-\sin ^{-1} x\right)\left(\sin ^{-1} x\right)}\right.}{\sqrt{1-x^2}} d x \quad \ldots . .\left[\because \sin ^{-1} x+\cos ^{-1} x=\frac{\pi}{2}\right] $
Put $\sin ^{-1} x=t$
$\therefore \frac{1}{\sqrt{1-x^2}} d x= dt$
When $x =\frac{1}{\sqrt{2}}, t =\frac{\pi}{4}$ and when $x =1, t =\frac{\pi}{2}$
$ \therefore I =\int_{\frac{\pi}{4}}^{\frac{\pi}{2}}\left( e ^{\frac{\pi}{2}- t }\right) t dt$
$=\left[ t \int e ^{\frac{\pi}{2}- t } dt \right]_{\frac{\pi}{4}}^{\frac{\pi}{2}}-\int_{\frac{\pi}{4}}^{\frac{\pi}{2}}\left[\frac{ d }{ dt }( t ) \int e ^{\frac{\pi}{2}- t } dt \right] dt$
$=\left[ t \cdot \frac{ e ^{\frac{\pi}{2}- t }}{-1}\right]_{\frac{\pi}{4}}^{\frac{\pi}{2}}-\int_{\frac{\pi}{4}}^{\frac{\pi}{2}} 1 \cdot \frac{ e ^{\frac{\pi}{2}- t }}{-1} dt$
$=-\left(\frac{\pi}{2} e ^0-\frac{\pi}{4} e ^{\frac{\pi}{4}}\right)+\int_{\frac{\pi}{4}}^{\frac{\pi}{2}} e ^{\frac{\pi}{2}- t } dt$
$=-\left(\frac{\pi}{2}-\frac{\pi}{4} e ^{\frac{\pi}{4}}\right)+\left[\frac{ e ^{\frac{\pi}{2}- t }}{-1}\right]_{\frac{\pi}{4}}^{\frac{\pi}{2}}$
$=-\frac{\pi}{2}+\frac{\pi}{4} e ^{\frac{\pi}{4}}-\left( e ^0- e ^{\frac{\pi}{4}}\right)$
$=-\frac{\pi}{2}+\frac{\pi}{4} e ^{\frac{\pi}{4}}-1+ e ^{\frac{\pi}{4}}$
$\therefore I = e ^{\frac{\pi}{4}}\left(\frac{\pi}{4}+1\right)-\left(\frac{\pi}{2}+1\right) $
$=\int_{\frac{1}{\sqrt{2}}}^1 \frac{\left( e ^{\left.\frac{\pi}{2}-\sin ^{-1} x\right)\left(\sin ^{-1} x\right)}\right.}{\sqrt{1-x^2}} d x \quad \ldots . .\left[\because \sin ^{-1} x+\cos ^{-1} x=\frac{\pi}{2}\right] $
Put $\sin ^{-1} x=t$
$\therefore \frac{1}{\sqrt{1-x^2}} d x= dt$
When $x =\frac{1}{\sqrt{2}}, t =\frac{\pi}{4}$ and when $x =1, t =\frac{\pi}{2}$
$ \therefore I =\int_{\frac{\pi}{4}}^{\frac{\pi}{2}}\left( e ^{\frac{\pi}{2}- t }\right) t dt$
$=\left[ t \int e ^{\frac{\pi}{2}- t } dt \right]_{\frac{\pi}{4}}^{\frac{\pi}{2}}-\int_{\frac{\pi}{4}}^{\frac{\pi}{2}}\left[\frac{ d }{ dt }( t ) \int e ^{\frac{\pi}{2}- t } dt \right] dt$
$=\left[ t \cdot \frac{ e ^{\frac{\pi}{2}- t }}{-1}\right]_{\frac{\pi}{4}}^{\frac{\pi}{2}}-\int_{\frac{\pi}{4}}^{\frac{\pi}{2}} 1 \cdot \frac{ e ^{\frac{\pi}{2}- t }}{-1} dt$
$=-\left(\frac{\pi}{2} e ^0-\frac{\pi}{4} e ^{\frac{\pi}{4}}\right)+\int_{\frac{\pi}{4}}^{\frac{\pi}{2}} e ^{\frac{\pi}{2}- t } dt$
$=-\left(\frac{\pi}{2}-\frac{\pi}{4} e ^{\frac{\pi}{4}}\right)+\left[\frac{ e ^{\frac{\pi}{2}- t }}{-1}\right]_{\frac{\pi}{4}}^{\frac{\pi}{2}}$
$=-\frac{\pi}{2}+\frac{\pi}{4} e ^{\frac{\pi}{4}}-\left( e ^0- e ^{\frac{\pi}{4}}\right)$
$=-\frac{\pi}{2}+\frac{\pi}{4} e ^{\frac{\pi}{4}}-1+ e ^{\frac{\pi}{4}}$
$\therefore I = e ^{\frac{\pi}{4}}\left(\frac{\pi}{4}+1\right)-\left(\frac{\pi}{2}+1\right) $
Need a full question paper?
Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.
Explore more
Similar questions
A fair coin is tossed 8 times. Find the probability that :
(i) it shows no head.
(ii) it shows head at least once.
(i) it shows no head.
(ii) it shows head at least once.
Write the following statements in symbolic form
Milk is white if and only if the sky is not blue
Find the equations of tangents and normals to the curve at the point on it.
→$x^2-\sqrt{3} x y+2 y^2=5$ at $(\sqrt{ } 3,2)$
Show that the combined equation of a pair of lines through the origin and each making an
→angle of $\alpha$ with the line $x+y=0$ is $x^2+2(\sec 2 \alpha) x y+y^2=0$.
Integrate the following w.r.t. x:
→$\log (1+\cos x)-x \tan \left(\frac{x}{2}\right)$
Find a unit vector perpendicular to the plane containing the point (a, 0, 0), (0, b, 0), and (0, 0, c). What is the area of the triangle with these vertices?
The following is the c.d.f. of a r.v. X:
Find
(i) p.m.f. of X
(ii) P( -1 ≤ X ≤ 2)
(iii) P(X ≤ X > 0).
Examine whether the following statement pattern is a tautology, a contradiction or a contingency.
(p ∧ ~ q) → (~ p ∧ ~ q)
$\int \frac{(2 \log x+3)}{x(3 \log x+2)\left[(\log x)^2+1\right]} d x$
→Verify Lagrange’s mean value theorem for the following functions: f(x) = (x – 1)(x – 2)(x – 3) on [0, 4]