Cambridge Additional Mathematics

430 Integration (Chapter 15)

Example 12 Self Tutor

Integrate with respect tox:

a 2 e^2 x¡e¡^3 x b 2 sin(3x) + cos(4x+¼)

a

Z

(2e^2 x¡e¡^3 x)dx

=2(^12 )e^2 x¡(¡^13 )e¡^3 x+c

=e^2 x+^13 e¡^3 x+c

b

Z

(2 sin(3x) + cos(4x+¼))dx

=¡^23 cos(3x)+^14 sin(4x+¼)+c

EXERCISE 15F

1 Find:

a

Z

(2x+5)^3 dx b

Z

1

(3¡ 2 x)^2

dx c

Z

4

(2x¡1)^4

dx

d

Z

(4x¡3)^7 dx e

Z p

3 x¡ 4 dx f

Z

10

p

1 ¡ 5 x

dx

g

Z

3(1¡x)^4 dx h

Z

4

p

3 ¡ 4 x

dx

2 Integrate with respect tox:

a sin(3x) b 2 cos(¡ 4 x)+1 c 3 cos

¡x

2

¢

d 3 sin(2x)¡e¡x e 2 sin

¡

2 x+¼ 6

¢

f ¡3 cos

¡¼

4 ¡x

¢

g cos(2x) + sin(2x) h 2 sin(3x) + 5 cos(4x) i^12 cos(8x)¡3 sinx

3 Find y=f(x) given

dy

dx

=

p

2 x¡ 7 and that y=11when x=8.

4 The function f(x) has gradient function f^0 (x)=

4

p

1 ¡x

, and the curve y=f(x) passes through

the point (¡ 3 ,¡11).

Find the point on the graph of y=f(x) withx-coordinate¡ 8.

5 Find:

a

Z

3(2x¡1)^2 dx b

Z

(x^2 ¡x)^2 dx c

Z

(1¡ 3 x)^3 dx

d

Z

(1¡x^2 )^2 dx e

Z

4

p

5 ¡xdx f

Z

(x^2 +1)^3 dx

6 Find:

a

Z ¡

2 ex+5e^2 x

¢

dx b

Z ¡

3 e^5 x¡^2

¢

dx c

Z ¡

e^7 ¡^3 x

¢

dx

d

Z

(ex+e¡x)^2 dx e

Z

(e¡x+2)^2 dx f

Z μ

x¡

5

(1¡x)^2

¶

dx

7 Find an expression forygiven that

dy

dx

=(1¡ex)^2 , and that the graph hasy-intercept 4.

=2£¡^13 cos(3x)+^14 sin(4x+¼)+c

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Additional Mathematics
Y:\HAESE\CAM4037\CamAdd_15\430CamAdd_15.cdr Monday, 7 April 2014 3:59:41 PM BRIAN