MATH1010 Assignment 5 Suggested Solution

1. (a) (b) (c)

MATH1010 Assignment 5 Suggested Solution

x

dx

1 - x2

Put x = sin y

sin y

=

cos y dy

1 - sin2 y

dx = cos y dy

sin y

=

cos y dy

cos y

= - cos y + C

= - 1 - x2 + C

x (1 + x2)2 dx

Put y = 1 + x2

1 = 2y2 dy

dy = 2x dx

1 =- +C

2y

1

=

- 2(1

+

x2)

+

C

ex dx

2 + ex

1

=

dy

2+y

Put y = ex dy = ex dx

= ln(2 + y) + C

= ln(2 + ex) + C

1

(d) 2. (a)

(b) 3. (a)

x x - 1 dx

= (y + 1) y dy

Put y = x - 1 dy = dx

3

1

= y 2 + y 2 dy

25 23

= y2 + y2 +C

53

2

52

3

= (x - 1) 2 + (x - 1) 2 + C

5

3

x cos(5 - x) dx = -x sin(5 - x) + sin(5 - x) dx = -x sin(5 - x) + cos(5 - x) + C

x2e-2x dx

= - 1 x2e-2x + xe-2x dx 2

= - 1 x2e-2x - 1 xe-2x + 1 e-2x dx

2

2

2

= - 1 x2e-2x - 1 xe-2x - 1 e-2x + C

2

2

4

sin5 x cos x dx

= sin5 x d sin x = 1 sin6 x + C

6

2

(b) cos 2x - cos 8x

sin 3x sin 5x = 2

sin 3x sin 5x dx

cos 2x - cos 8x

=

dx

2

sin 2x sin 8x

=

-

+C

4

16

(c)

sin2

x

=

1 (

-

cos

2x )

2

sin4 x dx

= ( 1 - cos 2x )2 dx

2

1 - 2 cos 2x + cos2 2x

=

dx

4

x sin 2x 1 + cos 4x

=-

+

dx

44

8

3x sin 2x sin 4x

=-

+

+C

8

4

32

(d)

cos5 x sin4 x dx

= cos4 x sin4 x d sin x

= (1 - sin2 x)2 sin4 x d sin x

sin5 x 2 sin7 x sin9 x

=

-

+

+C

5

7

9

3

(e) (f) 4. (a)

sin2 x cos4 x dx

1 - cos 2x 1 + cos 2x 2

=

dx

2

2

1 + cos 2x - cos2 2x - cos3 2x

=

dx

8

x sin 2x

cos2 2x

cos3 2x

=+

-

dx -

dx

8 16

8

8

x sin 2x 1 + cos 4x

1 - sin2 2x

=+

-

dx -

d sin 2x

8 16

16

16

x sin 4x sin3 2x

=-

+

+C

16 64

48

tan5 x dx

= tan3 x(sec2 x - 1) dx

= tan3 x sec2 x dx - tan3 x dx

tan4 x

=

-

tan x(sec2 x - 1) dx

4

tan4 x tan2 x

=

-

+ ln | sec x| + C

4

2

dx

(1

-

x2)

3 2

cos y

=

dy

cos3 y

= sec2 y dy

Put x = sin y dx = cos y dy

= tan y + C

x

=

+C

1 - x2

4

(b)

dx

(1

+

x2)

3 2

Put x = tan y

sec2 y

=

(1

+

tan2

y)

3 2

dy

dx = sec2 y dy

= cos y dy

= sin y + C

x

=

+C

1 + x2

(c)

x2

dx

9 - x2

9 sin2 y

=

3 cos y dy

3 cos y

Put x = 3 sin y dx = 3 cos y dy

= 9 sin2 y dy

9y 9 sin 2y

=-

+C

2

4

= 9 sin-1 x - x

9 - x2 +C

2

3

2

(d)

dx

Put x = 2 tan y

x2 x2 + 4

1 = 4 sin2 y sec y dy

dx = 2 sec2 y dy

1

= 4 sin2 y d sin y

1

=-

+C

4sin y

4 + x2

=-

+C

4x

5

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