Question

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

Answer 1

`f(x)=1/3 (x^2-2)^(3/2) +3-(7sqrt7)/3`

Explanation:

`f(x)=intxsqrt(x^2-2)dx=intx(x^2-2)^(1/2)dx`

Note that the x outside of the square root is part of the derivative of the inside so we can just integrate `(x^2-2)^(1/2)`

`f(x)=intx(x^2-2)^(1/2)dx = 2/3 (x^2-2)^(3/2)*1/2`-> since the 2 from the derivative of the inside is missing.

`f(x)=1/3 (x^2-2)^(3/2) +C`
`f(3)=1/3 (9-2)^(3/2)+C=3`
`(7sqrt7)/3 +C =3`
`C=3-(7sqrt7)/3`
`f(x)=1/3 (x^2-2)^(3/2) +3-(7sqrt7)/3`