Question

Integrate `intsqrt(9-x^2)"d"x`?

Answer 1

`intsqrt(9-x^2)"d"x=1/2(xsqrt(9-x^2)+9arcsin(x/3))+"c"`

Explanation:

`intsqrt(9-x^2)"d"x=intsqrt(9(1-(x/3)^2))"d"x=3intsqrt(1-(x/3)^2)"d"x`

Now let `sinu=x/3` and `cosu"d"u=1/3dx`

So

`3intsqrt(1-(x/3)^2)"d"x=9intcos^2u"d"u=9/2int1+cos2u"d"u`

`=9/2(u+1/2sin2u)+"c"=1/2(9u+9sinucosu)+"c"`

We now substitute back `u=arcsin(x/3)`

`intsqrt(9-x^2)"d"x=1/2(9arcsin(x/3)+xsqrt(9-x^2))+"c"`