Edexcel A-Level Mathematics A2 Fp3 4 Integration Questions

Practise exact FP3 integration with hyperbolic forms, inverse trig forms, substitutions, quadratic surds and curve-measure formulae.

Exam points

complete the square to match integrals with arcsin, arctan or logarithmic forms
form curve length or surface area integrals, then simplify to logs or exponential constants

Covered syllabus points

FP3.4.1 - Integration of hyperbolic functions FP3.4.2 - Integration of inverse trigonometric FP3.4.3 - Integration FP3.4.4 - Use of substitution for integrals FP3.4.5 - Reduction formulae FP3.4.6 - The

[Maximum number: 8]

In this question you must show all stages of your working. Solutions relying on calculator technology are not acceptable.

Figure 1

Figure 1 shows the curve with equation

y=\ln\left(\tanh\frac{x}{2}\right) \quad 1\leqslant x\leqslant 2

(a)

Show that the length, s, of the curve is given by

s=\int_1^2 \coth x\,\mathrm{d}x

[ 4 ]

\frac{\mathrm{d}y}{\mathrm{d}x}=\frac{1}{\tanh(x/2)}\cdot\frac12\operatorname{sech}^2(x/2)

=\frac{1}{2\sinh(x/2)\cosh(x/2)}=\operatorname{cosech}x

\sqrt{1+\left(\frac{\mathrm{d}y}{\mathrm{d}x}\right)^2}=\sqrt{1+\operatorname{cosech}^2x}

=\sqrt{\coth^2x}=\coth x\quad (1\le x\le2)

s=\int_1^2\coth x\,\mathrm{d}x

M1: obtains an appropriate expression for dy/dx.
M1: applies the arc length formula.
ddM1: uses identities to obtain 1+(dy/dx)^2 in terms of x.
A1*: reaches the printed integral with no errors.

(b)

Hence show that

s=\ln\left(e+\frac1e\right)

[ 4 ]

s=\int_1^2\coth x\,\mathrm{d}x

=\left[\ln(\sinh x)\right]_1^2=\ln(\sinh2)-\ln(\sinh1)

=\ln\left(\frac{\sinh2}{\sinh1}\right)

\sinh2=2\sinh1\cosh1\Rightarrow \frac{\sinh2}{\sinh1}=2\cosh1=e+e^{-1}

s=\ln\left(e+\frac1e\right)

B1: integrates coth x to ln(sinh x).
M1/dM1: substitutes limits and uses hyperbolic/exponential identities or log laws.
A1*: obtains the required result by algebraic integration.

[Maximum number: 6]

In this question you must show all stages of your working.

Solutions relying entirely on calculator technology are not acceptable.

Use the substitution $x=3\sec\theta$ to determine the exact value of

\int_{2\sqrt3}^{6}\frac{18}{(x^2-9)^{3/2}}\,dx

Give your answer in the form $A+B\sqrt3$ , where A and B are constants to be found.

x=6\Rightarrow \theta=\frac\pi3,\qquad x=2\sqrt3\Rightarrow \theta=\frac\pi6

Since $dx=3\sec\theta\tan\theta\,d\theta$ ,

\int\frac{18}{(x^2-9)^{3/2}}\,dx=\int\frac{54\sec\theta\tan\theta}{(9\sec^2\theta-9)^{3/2}}\,d\theta=2\int\frac{\cos\theta}{\sin^2\theta}\,d\theta

2\int\cosec\theta\cot\theta\,d\theta=-2\cosec\theta

[-2\cosec\theta]_{\pi/6}^{\pi/3}=4-\frac43\sqrt3

[Maximum number: 9]

I_n=\int_0^k x^n(k-x)^{\frac12}\,dx \quad n\geqslant 0

where k is a positive constant.

(a)

Show that

I_n=\frac{2kn}{3+2n}I_{n-1}\quad n\ge1

[ 5 ]

u=x^n,\quad \mathrm{d}v=(k-x)^{1/2}\,\mathrm{d}x,\quad v=-\frac23(k-x)^{3/2}

I_n=\left[-\frac23x^n(k-x)^{3/2}\right]_0^k+\frac{2n}{3}\int_0^k x^{n-1}(k-x)^{3/2}\,\mathrm{d}x

=\frac{2n}{3}\int_0^k x^{n-1}(k-x)(k-x)^{1/2}\,\mathrm{d}x

=\frac{2n}{3}\left(kI_{n-1}-I_n\right)

\left(1+\frac{2n}{3}\right)I_n=\frac{2kn}{3}I_{n-1}

I_n=\frac{2kn}{3+2n}I_{n-1}

M1/A1: integration by parts in the correct direction with correct expression.
dM1/ddM1: rewrites (k-x)^(3/2) and expresses the result in I_n and I_(n-1).
A1*: reaches the printed reduction formula.

(b)

Given that

\int_0^k x^2(k-x)^{1/2}\,\mathrm{d}x=\frac{9\sqrt3}{280}

use the result in part (a) to determine the exact value of k.

[ 4 ]

I_2=\frac{4k}{7}I_1,\quad I_1=\frac{2k}{5}I_0

I_0=\int_0^k(k-x)^{1/2}\,\mathrm{d}x=\left[-\frac23(k-x)^{3/2}\right]_0^k=\frac23k^{3/2}

I_2=\frac{4k}{7}\cdot\frac{2k}{5}\cdot\frac23k^{3/2}=\frac{16}{105}k^{7/2}

\frac{16}{105}k^{7/2}=\frac{9\sqrt3}{280}

k^{7/2}=\frac{27\sqrt3}{128}=\left(\frac34\right)^{7/2}

k=\frac34

M1: uses the reduction formula to express I_2 in terms of I_0.
M1: evaluates I_0.
ddM1: solves an equation of the form a k^(7/2)=9sqrt(3)/280.
A1: exact value k=3/4.

All question bank results loaded

Covered syllabus points

Question 1

Question (a)

Question (b)

Question 2

Question 3

Question (a)

Question (b)