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

Practise Edexcel IAL P4.6.2 integration by substitution and parts: transform integrals, update limits, and retain exact constants.

Exam points

Apply a given substitution, transform dx and update definite-integral limits
Use integration by parts correctly for products and repeat it when required
Evaluate the transformed integral exactly and retain constants or exact forms

[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