$\int$ (sin^-1 x^1/2 - cos^-1 x^1/2)/(sin^-1 x^1/2 +cos^-1 x^1/2) dx

gokusuper

$\int$ (sin^-1 x^1/2 - cos^-1 x^1/2)/(sin^-1 x^1/2 +cos^-1 x^1/2) dx

flowers_rsss

$\int \pi/2 - 2{cos}^{-1}\sqrt{x}/\pi/2$

$\int 1 - 4/\pi {cos}^{-1}\sqrt{x}$

$= x-4/\pi \int 2y{cos}^{-1}y where y =\sqrt{x}$

applying integration by parts we get

$x-8/\pi [ytany-log secy ]$ where y =root over x

correct if i am wrong!!

chinmay_saxena01

sin^-1x+cos^-1x=pie/2....so den. becomes pie/2...
so from question integration sin^-1rootx-cos^-1rootx/pie/2
=>2/pie integration of sin^-1rootx-cos^-1rootx...
=>2/pie integration of sin^-1rootx-(pie/2 -sin^-1rootx)
it becomes....integration of4/piesin^-1rootx-1.....i1-i2
i1 =integration of sin^-1rootx...now put x=t^2.....and integrate u will get 2/pie{root(x-x^2)-(1-2x)sin^-1rootx}...this is i1
and finally u will get
2/pie{root(x-x^2)-(1-2x)sin^-1rootx}-x+c

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(sin-1 x1/2 - cos-1 x1/2)/(sin-1 x1/2 +cos-1 x1/2) dx

$\int$ (sin^-1 x^1/2 - cos^-1 x^1/2)/(sin^-1 x^1/2 +cos^-1 x^1/2) dx