Question

Integrate the following with respect to x:

`(x sin^-1 x)/sqrt(1 - x^2)`

Answer 1

`int (x sin^-1 x)/sqrt(1 - x^2) * "d"x`

Put x = sin θ

⇒ dx = cos θ dθ

`int (x sin^-1x)/sqrt(1 - x^2)  * "d"x = int (sin theta sin^-1 (sin theta))/sqrt(1 - sin^2theta)  costheta  "d"theta`

= `int (theta sin theta)/sqrt(cos^2 theta) * cos theta  "d"theta`

=`int (theta sin theta)/cos theta * cos theta  "d"theta`

`int (x sin^-1x)/sqrt(1 - x^2)  * "d"x = int theta sin theta  "d"theta`   ..........(1)

Consider `int theta sin theta  "d"theta`

u = θ

u' = 1

u" = 0

dv = sin θ dθ

⇒ v = `int sin theta  "d"theta`

⇒ v = – cos θ

v₁ =  `int "v"  "d"theta`

= `int - cos theta  "d"theta`

= – sin θ

v₂ =  `int "v"_1  "d"theta`

= `int - sin theta  "d"theta`

= (– cos θ) 

= cos θ

`int "u"  "dv"` = uv – u'v₁ + u"v₂ – u"'v₃ + ...........

`int theta sin theta  = theta(- cos theta)- 1(- sin theta)  + 0(cos theta)`

= `- theta cos theta + sin theta + "c"`

(1) ⇒ `int (x sin^-1x)/sqrt(1 - x^2)  "d"x = - theta cos theta + sin theta + "c"`

x = sin θ

⇒ θ = `sin^-1x`

cos θ = `sqrt(1 - sin^2theta)`

= `sqrt(1 - x^2)`

∴ `int (x sin^-1x)/sqrt(1 - x^2)  "d"x = - sin^-1x (sqrt(1 - x^2)) + x + "c"`

 `int (x sin^-1x)/sqrt(1 - x^2)  "d"x = - sqrt(1 - x^2) sin^-1x + x + "c"`