Question

sin⁻¹√x/x+a ,Integrate the given function defined on proper domain w.r.t. x.

Answer 1

HELLO DEAR,

put x = atan²t so that dx = (2asec²t*tant)dt.

therefore,
$\bold{\int{sin^{-1}\sqrt{\frac{x}{a + x}}}\,dx}$

$\bold{= \int{sin^{-1}[\sqrt{\frac{atan^2t}{a(1 + tan^2t)}}]2asec^2t.tant}\,dt}$

$\bold{= 2a\int{t(sec^2t.tant)}\,dt}$

$\bold{= 2a[t.1/2tan^2t - \int{1.1/2tan^2t}\,dt]}$

[integrating by parts and using $\bold{\int{sec^2t.tant}\,dt = 1/2tan^2t}$]

= at(tan²t) - a$\bold{\int{(sec^2t - 1)}\,dt}$

= at(tan²t) - $\bold{a\int{sec^2t}\,dt + a\int{1}\,dt}$

= at(tan²t) - atant + at + C

$\bold{=a(tan^{-1}\sqrt{x \over a})*(x/a) - a \sqrt{x \over a} + atan^{-1}\sqrt{x \over a} + C}$

$\bold{=xtan^{-1}\sqrt{x \over a} - \sqrt{ax} + atan^{-1}\sqrt{x \over a} + C}$

I HOPE ITS HELP YOU DEAR,
THANKS