Question

What is `f(x) = int xsin3x dx` if `f(pi/3) = -2`?

Answer 1

`f(x) = -x/3cos(3x) + 1/9sin(3x) - pi/9`

Explanation:

Use integration by parts:

`intudv = uv - intv du`
`u = x; dv = sin(3x) dx`
`du = dx; intdv = intsin(3x) dx; v = -1/3 cos(3x)`
`intudv = -x/3cos(3x) + 1/3int cos(3x) du`

`f(x) = intudv = -x/3cos(3x) + 1/9sin(3x) + C`

To calculate the constant, C use the fact that: `f(pi/3) = -2;`
`f(pi/3) = -pi/9cos(pi) + cancel(1/3 sin(pi)) = -pi/9`

`f(x) = -x/3cos(3x) + 1/9sin(3x) - pi/9`