Question

Suppose that f(x) is continuous and differentiable,f(2)= -2, f(6) =-6, and the integral from 2 to 6 of f(x) dx equals -10. Compute the integral from 2 to 6 of x*f ' (x) dx

Answer 1

Use partial integration with U=x and dV=f'(x)dx, we get
\[ \int_{2}^{6}{xf'(x) \, dx} = xf(x)|_{x=2}^{6}-\int_{2}^{6}{f(x)dx} = \]
\[ = (6f(6)-2f(2))-(-10) = (-36+4)+10 = -22 \]