The Weierstrass substitutionEdexcel A-Level Further Maths: Mind map

Exam questions on The Weierstrass substitution

1. The substitution $t=\tan\frac{x}{2}$ is used to find integrals, for $0\le x<\pi$.
   Hence find the exact value of $\displaystyle\int_0^{\pi/2}\frac{1}{1+\cos x}\,\mathrm{d}x$.2 marks
2. The integral $I=\displaystyle\int_0^{\pi/2}\frac{1}{1+\sin x}\,\mathrm{d}x$ is evaluated using the substitution $t=\tan\frac{x}{2}$.
   Hence find the exact value of $I$.2 marks
3. Let $J=\displaystyle\int_{\pi/3}^{\pi/2}\frac{1}{1+\sin x-\cos x}\,\mathrm{d}x$, and let $t=\tan\frac{x}{2}$.
   Show that $J=\displaystyle\int_{1/\sqrt3}^{1}\frac{1}{t(1+t)}\,\mathrm{d}t$.3 marks