Evaluate the following integral

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

Problem

Evaluate the following integral, showing all necessary steps: (i) Show that

\displaystyle \int 2x\cos\bigl(x^{2}\bigr)\,dx = \sin\bigl(x^{2}\bigr)+C.

(ii) Evaluate

\displaystyle \int \frac{1}{x}\,\sin(\ln x)\,dx.

(iii) Hence, determine

\displaystyle \int \Bigl(2x\cos\bigl(x^{2}\bigr)-\tfrac{1}{x}\,\sin(\ln x)\Bigr)\,dx.