Question
Evaluate: $\int_0^{\frac{\pi}{4}} \log (1+\tan x) d x$
✓
Answer
$\text { Let } I =\int_0^\pi \frac{1}{3+2 \sin x+\cos x} d x$
Put $\tan \left(\frac{x}{2}\right)= t$
$\therefore x =2 \tan ^{-1} t$
$\therefore dx =\frac{2 dt }{1+ t ^2}, \sin x =\frac{2 t }{1+ t ^2}$ and $\cos x =\frac{1- t ^2}{1+ t ^2}$
When $x =0, t =0$ and when $x =\pi, t =\infty$
$ \therefore I =\int_0^{\infty} \frac{1}{3+2\left(\frac{2 t }{1+ t ^2}\right)+\frac{1- t ^2}{1+ t ^2}} \times \frac{2 dt }{1+ t ^2}$
$=\int_0^{\infty} \frac{2 dt }{3+3 t ^2+4 t +1- t ^2}$
$=\int_0^{\infty} \frac{2 dt }{2 t ^2+4 t +4}$
$=\int_0^{\infty} \frac{ dt }{ t ^2+2 t +2} $
$=\int_0^{\infty} \frac{ dt }{ t ^2+2 t +1+1}$
$=\int_0^{\infty} \frac{ dt }{( t +1)^2+1^2}$
$=\left[\tan ^{-1}( t +1)\right]_0^{\infty}$
$=\tan ^{-1}(1+\infty)-\tan ^{-1}(1+0)$
$=\tan ^{-1}(\infty)-\tan ^{-1}(1)$
$=\frac{\pi}{2}-\frac{\pi}{4}$
$\therefore I ==\frac{\pi}{4}$
Put $\tan \left(\frac{x}{2}\right)= t$
$\therefore x =2 \tan ^{-1} t$
$\therefore dx =\frac{2 dt }{1+ t ^2}, \sin x =\frac{2 t }{1+ t ^2}$ and $\cos x =\frac{1- t ^2}{1+ t ^2}$
When $x =0, t =0$ and when $x =\pi, t =\infty$
$ \therefore I =\int_0^{\infty} \frac{1}{3+2\left(\frac{2 t }{1+ t ^2}\right)+\frac{1- t ^2}{1+ t ^2}} \times \frac{2 dt }{1+ t ^2}$
$=\int_0^{\infty} \frac{2 dt }{3+3 t ^2+4 t +1- t ^2}$
$=\int_0^{\infty} \frac{2 dt }{2 t ^2+4 t +4}$
$=\int_0^{\infty} \frac{ dt }{ t ^2+2 t +2} $
$=\int_0^{\infty} \frac{ dt }{ t ^2+2 t +1+1}$
$=\int_0^{\infty} \frac{ dt }{( t +1)^2+1^2}$
$=\left[\tan ^{-1}( t +1)\right]_0^{\infty}$
$=\tan ^{-1}(1+\infty)-\tan ^{-1}(1+0)$
$=\tan ^{-1}(\infty)-\tan ^{-1}(1)$
$=\frac{\pi}{2}-\frac{\pi}{4}$
$\therefore I ==\frac{\pi}{4}$
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
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).
Show that $\frac{d y}{d x}=\frac{y}{x}$ in the following, where a and $p$ are constants.
→$e^{\frac{x^7-y^7}{x^7+y^7}}=a$
Solve the equations $2x + 5y = 1 $ and $3x + 2y = 7$ by the method of inversion
→Assume that a spherical raindrop evaporates at a rate proportional to its surface area. If its radius originally is 3 mm and 1 hour later has been reduced to 2 mm, find an expression for the radius of the raindrop at any time t.
Are the four points $A(1,-1,1), B(-1,1,1), C(1,1,1)$ and $D(2,-3,4)$ coplanar? Justify your
→Solve the following equations by inversion method $:2x + 6y = 8, x + 3y = 5$
→An open box is to be cut out of piece of square card board of side $18\ cm$ by cutting of equal squares from the corners and turning up the sides. Find the maximum volume of the box
→Find the volume of the parallelepiped spanned by the diagonals of the three faces of a cube of side a that meet at one vertex of the cube.
Differentiate the following w. r. t. x.$x^a+x^x+a^x$
→If one of the lines given by $a x^2+2 h x y+b y^2=0$ is perpendicular to $p x+q y=0$ then showthat $a p^2+2 h p q+b q^2=0$
→