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

1 Answer
Dec 11, 2017

#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#