Consider the definite integral

MATHEMATICS · Multi-part · A Level · Mathematics · Integration · Integration

Problem

J=\displaystyle \int_{0}^{\pi} x\sin(3x)\,dx.

(a) Determine an antiderivative of

x\sin(3x)

. (b) Evaluate

J

exactly. (c) Describe how the parity (odd or even symmetry) of the sine function combines with the linear factor

x

over the interval

[0,\pi]

to influence the sign of your answer in part (b).