Question

How do you integrate? I try again and again.

`int x^10sqrt(x-2)dx`

Answer 1

Use the substitution `x-2=u^2`.

Explanation:

Let

`I=intx^10sqrt(x-2)dx`

Apply the substitution `x-2=u^2`

`I=2int(u^2+2)^10u^2du`

Apply binomial expansion:

`I=2int{sum_(n=0)^10((10),(n))u^(2n)2^(10-n)}u^2du`

Rearrange:

`I=sum_(n=0)^10((10),(n))2^(11-n)intu^(2n+2)du`

Integrate directly:

`I=sum_(n=0)^10((10),(n))2^(11-n)/(2n+3)u^(2n+3)+C`

Reverse the substitution:

`I=sum_(n=0)^10((10),(n))2^(11-n)/(2n+3)(x-2)^((2n+3)/2)+C`