CAIE A-Level Mathematics 3.5.6 Integration by substitution

Cambridge International AS & A Level Mathematics 9709 syllabus for exams in 2028, 2029 and 20302028–2030

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

How this is tested

rewrite the whole integrand in u before integrating powers or trig-transformed forms
use substituted bounds to find exact areas bounded by curves, axes and vertical lines

(a)

Use the substitution $u=x^{2}-3$ to show that

\int_{\sqrt{7}}^{\sqrt{12}} \frac{4 x^{3}}{\sqrt{x^{2}-3}} \mathrm{~d} x=\int_{a}^{b} \frac{2(u+3)}{\sqrt{u}} \mathrm{~d} u,

where a and b are values to be found.

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(b)

Hence, find the exact value of $\int_{\sqrt{7}}^{\sqrt{12}} \frac{4 x^{3}}{\sqrt{x^{2}-3}} \mathrm{~d} x$ .

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