Practice section 10 of 10

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]

(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]

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