# CIE A level -Pure Mathematics 3 : Topic : 3.5 Integration – extend the idea of ‘reverse differentiation’ : Exam Style Questions Paper 3

### Question

(i) By differentiating $$\frac{1}{cosx}$$ , show that if y = sec x then$$\frac{dy}{dx}$$ = sec x tan x.

(ii) Show that $$\frac{1}{(secx-tanx)}$$ ≡ sec x + tan x.

(iii) Deduce that $$\frac{1}{(secx-tanx)^{2}}$$ ≡ $$2 sec^{2}$$
x − 1 + 2 sec x tan x.

(iv) Hence show that $$\int_{0}^{\frac{1}{4}\pi }\frac{1}{(secx-tanx)^{2}}dx$$=$$\frac{1}{4}(8\sqrt{2}-\pi )$$

(i) Use correct quotient or chain rule Obtain the given answer correctly having shown sufficient working
(ii) Use a valid method, e.g. multiply numerator and denominator by sec x + tan x, and a version of Pythagoras to justify the given identity
(iii) Substitute, expand
(sec x +$$tan x)^{2}$$
and use Pythagoras once
Obtain given identity
(iv) Obtain integral 2 tan x – x + 2 sec x
Use correct limits correctly in an expression of the form a tan x + bx + c sec x, or
equivalent, where abc 0
Obtain the given answer correctly

### Question

(i) Prove the identity tan $$tan2\Theta -tan\Theta sec2\Theta$$

(ii) Hence show that $$\int_{0}^{\frac{1}{6}\pi }tan\Theta sec2\Theta d\Theta =\frac{1}{2}In\frac{3}{2}$$

(i) EITHER: Use tan 2A formula to express LHS in terms of tanθ Express as a single fraction in any correct form Use Pythagoras or cos 2A formula Obtain the given result correctly

OR: Express LHS in terms of sin 2θ, cos 2θ, sin θ and cosθ
Express as a single fraction in any correct form
Use Pythagoras or cos 2A formula or sin(A – B) formula Obtain the given result correctly

(ii) Integrate and obtain a term of the form aln(cos2 ) θ or bln(cos ) θ (or secant equivalents)

Obtain integral $$-\frac{1}{2}$$ ln(cos 2 θ ) ln(cos θ )   , or equivalent

Substitute limits correctly (expect to see use of both limits)
Obtain the given answer following full and correct working

### Question

Let $$f(x)=\frac{\cos x}{1+\sin x}$$

(a) Show that f′x < 0 for all x in the interval $$-\frac{1}{2}\pi < x< \frac{3}{2}\pi$$

(b) Find $$\int_{\frac{1}{6}\pi }^{\frac{1}{2}\pi }f(x)dx$$. Give your answer in a simplified exact form.

Ans

(a) Use quotient or product rule
Obtain derivative in any correct form e.g $$\frac{-\sin x(1+\sin x)-\cos x(\cos x)}{(1+\sin x)^{2}}$$

Use Pythagoras to simplify the derivative
Justify the given statement

(b) State integral of the form a ln (1 + sin x)
State correct integral ln (1 + sin x) A1
Use limits correctly

Obtain answer $$ln\frac{4}{3}$$

### Question

Find $$\int_{\frac{1}{6}\pi}^{\frac{1}{3}\pi}x sec^2xdx$$. Give your answer in a simplified exact form.

Ans:

Integrate by parts and reach axtanx + b$$\int$$tan x dx
Obtain x tan x – $$\int$$ tan x dx
Complete the integration, obtaining a term $$\pm In cosx$$, or equivalent
Obtain integral xtan x + In cos x, or equivalent
Substitute limits correctly, having integrated twice
Use a law of logarithms
Obtain answer $$\frac{5}{18} \sqrt{3 \pi} – \frac{1}{2} In 3$$, or exact simplified equivalent

### Question

(a) Prove that  $$\frac{1-\cos 2\theta }{1+\cos 2\theta }=\tan ^{2}\theta$$                                                                               [2]

(b) Hence find the exact value of $$\int_{\frac{1}{6}\pi }^{\frac{1}{3}\pi } \frac{1-\cos 2\theta }{1+\cos 2\theta }$$    [4]

Ans

4 (a) Use correct double angle formula or t-substitution twice

Obtain $$\frac{1-\cos 2\theta }{1+\cos 2\theta }=tan^{2}\theta$$  from correct working

4 (b) Express 2 tanθ in terms of 2 secθ

Integrate and obtain terms tanθ – θ

Substitute limits correctly in an integral of the form a  tanθ + bθ  , where ab≠0

Obtain answer $$\frac{2}{3}\sqrt{3}-\frac{1}{6}\pi$$

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