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

1 Answer
Jan 23, 2016

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