CAIE A-Level Mathematics A2 3.5.6 Integration By Substitution Questions

Practise using a given substitution to rewrite integrals, adjust limits and evaluate exact values or areas under curves with clear working.

Exam points

Use a given substitution to rewrite integrals, adjust limits and evaluate exact values or areas under curves.

[Maximum number: 7]

Use the substitution $u=1-\sin x$ to find the exact value of

\int_{\pi}^{\frac{3}{2} \pi} \frac{\sin 2 x}{\sqrt{1-\sin x}} \mathrm{~d} x

Give your answer in the form $a+b \sqrt{2}$ where a and b are rational numbers to be determined.

State or imply $\mathrm{d} u=-\cos x \mathrm{~d} x$
Use $\sin 2 x=2 \sin x \cos x$ and write the integral in terms of u

*M1

Obtain $\pm 2 \int \frac{(1-u)}{\sqrt{u}} \mathrm{~d} u$ or equivalent
Integrate correctly to obtain $a u^{\frac{1}{2}}+b u^{\frac{3}{2}}$
Obtain correct $-4 u^{\frac{1}{2}}+\frac{4}{3} u^{\frac{3}{2}}$
Correctly use limits u=2 and 0 in an expression of the form $a u^{\frac{1}{2}}+b u^{\frac{3}{2}}$ OR limits $x=\frac{3}{2} \pi$ and $\frac{1}{2} \pi$ in an expression of the form
$a(1-\sin x)^{\frac{1}{2}}+b(1-\sin x)^{\frac{3}{2}}$
Obtain $\frac{8}{3}-\frac{4}{3} \sqrt{2}$

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CAIE A-Level Mathematics A2 3.5.6 Integration By Substitution Questions question 1