Chaos

In: Science

Submitted By huajui
Words 6354
Pages 26
CHAPTER 9 INTEGRATION

Focus on Exam 9
1



3 + sin x dx = cos2 x

∫cos

3
2

x

+ sin2 x dx cos x



= ∫[3 sec2 x + (cos x)−2 sin x] dx = ∫[3 sec2 x − (cos x)−2 (−sin x)] dx (cos x)−1 = 3 tan x − +c −1 = 3 tan x + 1 + c cos x 2 ∫sin2 x cos2 x dx = ∫(sin x cos x)2 dx = = = = 1 ∫2 sin 2x
2

dx

sin 2x = 2 sin x cos x 1 sin x cos x = sin 2x 2

∫ 4 sin
1 4

1

2

2x dx

∫

1 − cos 4x dx 2



cos 4x = 1 − 2 sin2 2x sin2 2x 1 − cos 4x = 2

∫ 1 1 = x − sin 4x + c 8 4
3

1 (1 − cos 4x) dx 8



du x 2 dx = 1 − u2 1 − x4



= = =

1 du 1 − u2 2



u = x2 du = 2x dx du x dx = 2

1 1 1+u × ln +c 2 2 1−u





1 1 + x2 ln +c 1 − x2 4





∫ a − x = − 2a dx 1
2 2

ln

a − x + c
© Oxford Fajar Sdn. Bhd. (008974-T) 2012

a+x

2

ACE AHEAD Mathematics (T) Second Term

4 Let x = sin θ dx = cos θ dθ dx = cos θ dθ

∫x

2

dx cos θ dθ = 2 2 1−x sin θ 1 − sin2 θ



= =

∫ sin θ ∫ sin dθ 2

cos θ dθ 2 cos2 θ

θ q 1

x

= ∫cosec2 θ dθ = − cot θ + c 1 − x2 =− x +c 5 Let u = ln x du 1 = dx x dx = du x dx 1 dx ∫ x ln x = ∫  x ln x =

1 − x2

[Shown]

∫u

du

= ln |u| + c = ln |ln x| + c 6 d (tan3 x) = 3 tan2 x sec2 x dx = 3(sec2 x − 1) sec2 x = (3 sec2 x − 3) sec2 x = 3 sec4 x − 3 sec2 x
4 2 3

[Shown]

∫(3 sec x − 3 sec x) dx = tan x + c ∫3 sec4 x dx − ∫3 sec2 x dx = tan3 x + c ∫3 sec4 x dx = ∫3 sec2 x dx + tan3 x + c = 3 tan x + tan3 x + c 1 c ∫ sec4 x dx = tan x + tan3 x + 3 3 1 ∫sec4 x dx = tan x + tan3 x + c′ 3

© Oxford Fajar Sdn. Bhd. (008974-T) 2012

Fully Worked Solution

3

7 ∫x sin x cos x dx = x =

1 ∫ 2 sin 2x dx 1

∫ 2 x sin 2x dx 1 1 1 1 = ∫ cos 2x   x − ∫ − cos 2x    dx 2 2 2 2
1 1 = − x cos 2x + cos 2x dx 4 4



1 1 = − x cos 2x + sin 2x + c 8 4 8



x 1 dx = 2 2 0 1 + x 1 = 2 1 = 2 1 = 2 1 =…...

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