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Integration — WFM03 IAL Further Mathematics practice questions · Study-IAL

Subjects
Further Mathematics
WFM03
3.4

Integration

WFM03Topic 3.43 questions

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Standard integrals, reduction formulae, arc length and surface area of revolution.

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Questions

Multiple choice: 1
Calculation: 2
Total: 3

Previous topic3.3Differentiation Next topic 3.5Vectors

Sample questions

Original practice questions written in the style of the exam.

Question 14 marks
Find the exact length of the arc of the curve $y = \cosh x$ from $x = 0$ to $x = \ln 2$ .
Mark scheme
- M1 $s = \displaystyle\int_0^{\ln 2} \sqrt{1 + \left(\frac{dy}{dx}\right)^2}\,dx$ with (1)
Question 26 marks
$I_n = \int_0^1 x^n e^x\,dx, \qquad n \geq 0$

1+(dxdy)2

d

x

\dfrac{dy}{dx} = \sinh x

A1

\sqrt{1 + \sinh^2 x} = \cosh x

(1)

M1

s = \left[\sinh x\right]_0^{\ln 2} = \sinh(\ln 2)

(1)

A1

= \dfrac{2 - \tfrac{1}{2}}{2} = \dfrac{3}{4}

(1)

xn

ex

d

x

,

n

≥

0

(a) Show that $I_n = e - nI_{n-1}$ for $n \geq 1$ .

(b) Hence find the exact value of $I_3$ .

Mark scheme

M1 Integrates by parts with $u = x^n$ , $dv = e^x\,dx$ (1)
A1 $I_n = \left[x^n e^x\right]_0^1 - n\displaystyle\int_0^1 x^{n-1}e^x\,dx$ (1)
A1 $= e - nI_{n-1}$ (1)
B1 $I_0 = e - 1$ (1)
M1 $I_1 = 1$ , $I_2 = e - 2$ (1)
A1 $I_3 = e - 3(e - 2) = 6 - 2e$ (1)

x

]

01

−

n∫01xn−1exdx

2