Pearson Edexcel IAL Mathematics P4.6.2 Integration by substitution & parts

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 $x=\frac{u+1}{2}$ .

B1

$\int_1^5\frac{3x}{\sqrt{2x-1}}\,dx =\int_1^9\left(\frac34u^{1/2}+\frac34u^{-1/2}\right)\,du$

M1 dM1 A1

$=\left[\frac12u^{3/2}+\frac32u^{1/2}\right]_1^9$

M1

$=\left(\frac12\cdot9^{3/2}+\frac32\cdot9^{1/2}\right) -\left(\frac12+\frac32\right)=16$

A1

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P4.6.2 - Integration by substitution and parts question 1