A-level · Integration

Integration by substitutionExam-style questions

Integrating by changing the variable — for integrals recognition cannot do. Choosing u to rename the awkward part; replacing dx with du; making every x disappear, including the strays; integrating in u and converting back; changing the limits of a definite integral so nothing converts back; and choosing a substitution, including a root or a trig function.

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32 questions · A-level level · mark scheme and worked solution on each · no sign-up to try

DifficultyAll Level 2 Level 3 Level 4 Level 5

1
The curve $C$ has equation

$y = \dfrac{x^2}{\sqrt{4 - x^2}}$ , $0 \le x < 2$
(a) Use the substitution $x = 2\sin\theta$ to show that

$\displaystyle\int_0^{\sqrt{3}} \dfrac{x^2}{\sqrt{4 - x^2}}\,\mathrm{d}x = \dfrac{2\pi}{3} - \dfrac{\sqrt{3}}{2}$ (6 marks)
(b) The line $y = \sqrt{3}\,x$ meets $C$ at the origin and at the point $P$ .

Find the exact area of the region enclosed by $C$ and the line. (4 marks)
Level 5 · 10 marksTry it
2
(a) Using the substitution $u = 1 + \sin x$ , find the exact value of $\displaystyle\int_{0}^{\frac{\pi}{2}}\dfrac{\cos x}{(1+\sin x)^{3}}\,dx$ . (4 marks)
(b) Find the exact value of $\displaystyle\int_{0}^{1} x\,\mathrm{e}^{-2x}\,dx$ . (4 marks)
Level 5 · 8 marksTry it
3
(a) Use the substitution $u = 2x - 3$ to find

$\displaystyle\int x\sqrt{2x - 3}\; dx$ ,

giving your answer in terms of $x$ . (4 marks)

The finite region $R$ is bounded by the curve $y = x\sqrt{2x - 3}$ , the $x$ -axis and the lines $x = 2$ and $x = 6$ .(b) Hence find the exact area of $R$ . (3 marks)
(c) Use the trapezium rule, with four strips of equal width, to estimate the area of $R$ . Explain, with reference to the shape of the curve, whether your estimate is an over-estimate or an under-estimate, and confirm that this is consistent with your answer to (b). (3 marks)
Level 5 · 10 marksTry it
4
Figure 3 shows the curve with equation

$y = \dfrac{x}{\sqrt{2x + 1}}, \qquad x \geqslant 0$

The region $R$ , shown shaded, is bounded by the curve, the $x$ -axis and the line $x = 4$ .Figure 3

Using the substitution $u = \sqrt{2x + 1}$ , find the exact area of $R$ . (9 marks)
Level 5 · 9 marksTry it
5
The curve $C$ has equation

$y = \dfrac{x}{\sqrt{2x + 1}}, \qquad x \geqslant 0$

The point $A$ on $C$ has $x$ -coordinate $4$ , and $O$ is the origin.

The finite region $R$ is bounded by $C$ and the line segment $OA$ .

Using the substitution $u = \sqrt{2x + 1}$ , find the exact area of $R$ . (9 marks)
Level 5 · 9 marksTry it

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