Question

`\intx^2\sqrt(9-x^2)dx`?

Symbolab does it a certain way, but is there a different method of approach?

Answer 1

`intx^2sqrt(9-x^2)dx=81/8[arcsin(x/3)-1/4sin(4arcsin(x/3))]`

Explanation:

I also follow similar way but my steps may be slightly more explained and are as follows.

Let `x=3sinu` then `dx=3cosudu` and this means `u=arcsin(x/3)`

and `intx^2sqrt(9-x^2)dx=int9sin^2usqrt(9-9sin^2u)*3cosudu`

= `81intsin^2ucos^2udu`

= `81/4intsin^2 2udu`

= `81/4int(1-cos4u)/2du`

= `81/8[intdu-intcos4udu]`

= `81/8[u-(sin4u)/4]`

= `81/8[arcsin(x/3)-1/4sin(4arcsin(x/3))]`