Question

What is `f(x) = int xsqrt(x-2) dx` if `f(2) = 3`?

Answer 1

`f(x)=(2x(x-2)^(3/2))/3-(4(x-2)^(5/2))/15+3`

Explanation:

`f(x)=int\ xsqrt(x-2)\ dx`

According to integration by parts,
`int\ u\ dv=uv-int\ v\ du`.

Set `u=x` and `dv=sqrt(x-2)\ dx=(x-2)^(1/2)\ dx`. Then, `du=dx` and `v=(2(x-2)^(3/2))/3`.

Thus,
`f(x)`
`=int\ xsqrt(x-2)\ dx`
`=(2x(x-2)^(3/2))/3-int\ (2(x-2)^(3/2))/3\ dx`
`=(2x(x-2)^(3/2))/3-(4(x-2)^(5/2))/15+C`

Now, since `f(2)=3`,
`(2*2(2-2)^(3/2))/3-(4(2-2)^(5/2))/15+C=3`
`C=3`

Thus, the final answer is `f(x)=(2x(x-2)^(3/2))/3-(4(x-2)^(5/2))/15+3`