Review practice · Section 3 of 3

Find $\displaystyle\int\dfrac{x+2}{\sqrt{1-8x-4x^2}}\,dx$.4 marks

(i) Express $\dfrac{4+x}{(1-x)(4+x^2)}$ in partial fractions.2 marks

(ii) Hence find $\displaystyle\int\dfrac{4+x}{(1-x)(4+x^2)}\,dx$.2 marks

Find $\displaystyle\int e^{2x}\cos4x\,dx$.4 marks

The use of a graphing calculator is not allowed in this question.

(i) By using integration by parts, find $\displaystyle\int x\tan^{-1}x\,dx$.4 marks

(ii) Find $\dfrac d{d\theta}(\tan^2\theta)$. Hence use the substitution $\theta=\tan^{-1}x$ to find $\displaystyle\int x\tan^{-1}x\,dx$.6 marks

(iii) Show algebraically that your answers to parts (i) and (ii) differ by a constant.1 mark