For a positive parameter

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

Problem

a

, define

I(a)=\displaystyle \int_{0}^{\sqrt{a}} 2x\,\sin\bigl(x^{2}+a\bigr)\,dx.

(i) Evaluate

I(a)

in terms of

a

. Clearly show each step of your working. (ii) Using the expression obtained in part (i), find the exact value of

I\!\bigl(\tfrac{\pi}{2}\bigr)