Question

What is `f(x) = int xsqrt(3-x) dx` if `f(3) = 0`?

Answer 1

`f(x) = -2/5(3-x)(x+2)sqrt(3-x)`

Explanation:

First we solve the indefinite integral:

`int xsqrt(3-x)dx = int (x-3+3) sqrt(3-x) dx = - int (3-x)^(3/2)dx +3 int(3-x)^(1/2)dx=2/5(3-x)^(5/2)-2(3-x)^(3/2)+C`

Now we see that:

`f(3) = C`, so `f(3) = 0 => C=0` and finally:

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