Edexcel A-Level Mathematics A2 P4.6.2 Integration By Substitution and Parts Questions

Practise choosing substitution or parts to transform definite and indefinite integrals, keeping exact forms and changed limits clear.

Exam points

transform definite integrals with a given substitution, updating dx and the limits

[Maximum number: 6]

Using a suitable substitution, find, using calculus, the value of

\int_{1}^{5} \frac{3 x}{\sqrt{2 x-1}} \mathrm{~d} x

(Solutions relying entirely on calculator technology are not acceptable.)

Use u=2x-1, so $dx=\frac12\,du$ and $x=\frac{u+1}{2}$ .
$\int_1^5\frac{3x}{\sqrt{2x-1}}\,dx =\int_1^9\left(\frac34u^{1/2}+\frac34u^{-1/2}\right)\,du$ $=\left[\frac12u^{3/2}+\frac32u^{1/2}\right]_1^9$ $=\left(\frac12\cdot9^{3/2}+\frac32\cdot9^{1/2}\right) -\left(\frac12+\frac32\right)=16$

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Edexcel A-Level Mathematics A2 P4.6.2 Integration By Substitution and Parts Questions question 1