The Weierstrass substitutionEdexcel A-Level Further Maths: Subtopic test

1. 1
   The substitution $t=\tan\frac{x}{2}$ is used to find integrals, for $0\le x<\pi$.

   (a)

   Which expression is equal to $\mathrm{d}x$ in terms of $t$ and $\mathrm{d}t$?

   [1 mark]

   • A$\dfrac{2\,\mathrm{d}t}{1+t^2}$
   • B$\dfrac{\mathrm{d}t}{1+t^2}$
   • C$\dfrac{(1+t^2)\,\mathrm{d}t}{2}$
   • D$\dfrac{2\,\mathrm{d}t}{1-t^2}$

   (b)

   The integral $\displaystyle\int\frac{1}{1+\cos x}\,\mathrm{d}x$ becomes which integral in $t$?

   [1 mark]

   • A$\displaystyle\int(1+t^2)\,\mathrm{d}t$
   • B$\displaystyle\int\mathrm{d}t$
   • C$\displaystyle\int\frac12\,\mathrm{d}t$
   • D$\displaystyle\int\frac{2}{1+t^2}\,\mathrm{d}t$

   (c)

   Hence find the exact value of $\displaystyle\int_0^{\pi/2}\frac{1}{1+\cos x}\,\mathrm{d}x$.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   The integral $I=\displaystyle\int_0^{\pi/2}\frac{1}{1+\sin x}\,\mathrm{d}x$ is evaluated using the substitution $t=\tan\frac{x}{2}$.

   (a)

   After the substitution, the integrand becomes which of these (with $\mathrm{d}t$)?

   [1 mark]

   • A$\dfrac{2}{(1+t)^2}$
   • B$\dfrac{2}{1+t^2}$
   • C$\dfrac{1}{(1+t)^2}$
   • D$\dfrac{2}{1+t}$

   (b)

   What is the upper limit of the integral in terms of $t$?

   [1 mark]

   • A$\frac{\pi}{4}$
   • B$\frac12$
   • C$\infty$
   • D$1$

   (c)

   Hence find the exact value of $I$.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   Let $J=\displaystyle\int_{\pi/3}^{\pi/2}\frac{1}{1+\sin x-\cos x}\,\mathrm{d}x$, and let $t=\tan\frac{x}{2}$.

   (a)

   Show that $J=\displaystyle\int_{1/\sqrt3}^{1}\frac{1}{t(1+t)}\,\mathrm{d}t$.

   [3 marks]

   (b)

   Hence find the exact value of $J$, giving your answer as a single logarithm.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   Let $t=\tan\frac{x}{2}$.

   (a)

   (i) Show that $\displaystyle\int\operatorname{cosec}x\,\mathrm{d}x=\ln\left|\tan\frac{x}{2}\right|+c$.
   (ii) Hence find the exact value of $\displaystyle\int_{\pi/3}^{\pi/2}\operatorname{cosec}x\,\mathrm{d}x$.

   [6 marks]

   (b)

   Use the substitution to find the exact value of $\displaystyle\int_0^{\pi/2}\frac{1}{2+\cos x}\,\mathrm{d}x$.

   [6 marks]

   Total for question 4: 12 marks

End of questions