Question

`intsin^-1(3x)dx`?

Answer 1

`intsin^-1(3x)dx=xsin^-1(3x)+1/3sqrt(1-9x^2)+C`

Explanation:

.

`intsin^-1(3x)dx`

Let `z=sin^-1(3x)`

`sinz=3x`, then `coszdz=3dx`, and `dx=1/3coszdz`

`intsin^-1(3x)dx=intz1/3coszdz=1/3intzcoszdz=1/3I`

Since we ended up with a product of two functions we will use integration by parts:

`u=z`, `dv=coszdz`

`du=dz`, `v=sinz`

`intudv=uv-intvdu`

`I=zsinz-intsinzdz=zsinz+cosz+C=`

Since `sinz=3x`, then `cosz=sqrt(1-sin^2z)=sqrt(1-9x^2)`

`I=3xsin^-1(3x)+sqrt(1-9x^2)+C`

Therefore,

`intsin^-1(3x)dx=xsin^-1(3x)+1/3sqrt(1-9x^2)+C`