The Weierstrass substitutionEdexcel A-Level Further Maths: Flashcards

What these 14 flashcards ask

What is the Weierstrass substitution?
State \sin x, \cos x in terms of t.
State dx in terms of t.
Why is dx=\frac{2\,dt}{1+t^2}?
Write \operatorname{cosec}x in terms of t.
\displaystyle\int\operatorname{cosec}x\,dx?
\displaystyle\int\frac{1}{1+\cos x}\,dx?
1+\cos x and 1+\sin x in terms of t?
What is 1+\sin x-\cos x in terms of t?
How do the limits change in a definite integral?
\displaystyle\int0^{\pi/2}\frac{1}{1+\sin x}\,dx?
\displaystyle\int{\pi/3}^{\pi/2}\frac{1}{1+\sin x-\cos x}\,dx?
Standard result for \int\frac{1}{a^2+t^2}\,dt?
When can you use this substitution?

Exam questions on The Weierstrass substitution

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
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
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

Written by the Exaim team, led by Shaun Daswani (Head of Upper Secondary, Improve ME Institute; MSc Financial Mathematics, Imperial College London; BSc, UCL) and Jason Daswani (operational lead, Improve ME Institute; LSE).