Reverse reciprocal-substitution bounds and reuse an integral

(a)(i) For $a>1$, use $u=1/x$ to show $\int_1^a dx/(1+x^2)=\int_{1/a}^1du/(1+u^2)$.4 marks

(ii) Hence show $\tan^{-1}a+\tan^{-1}(1/a)=k$, finding $k$.3 marks

(b)(i) Show $\int_0^1x^3/(1+x^2)\,dx=1/2-(1/2)\ln2$.2 marks

(ii) Without using a calculator, evaluate $\int_0^1x^2\tan^{-1}x\,dx$.3 marks