Cambridge Additional Mathematics

Integration (Chapter 15) 435

y

x

f(x) = 4 - x 2

O 2

6 Find the values ofbsuch that

Zb

0

cosxdx=

1

p

2

, 0 <b<¼.

7 Findyif:

a

dy

dx

=(x^2 ¡1)^2 b

dy

dx

= 400¡ 20 e

¡x 2

8 A curve y=f(x) has f^00 (x)=18x+10. Find f(x) if f(0) =¡ 1 and f(1) = 13.

9 If

Za

0

e^1 ¡^2 xdx=

e

4

, findain the form lnk.

10 Suppose f^00 (x)=3x^2 +2x and f(0) =f(2) = 3. Find:

a f(x) b the equation of the normal to y=f(x) at x=2.

11 a Find (ex+2)^3 using the binomial expansion.

b Hence find the exact value of

Z 1

0

(ex+2)^3 dx.

Review set 15B

1aUsefourupper and lower rectangles to find rational

numbersAandBsuch that:

A<

Z 2

0

(4¡x^2 )dx < B.

b Hence, find a good estimate for

Z 2

0

(4¡x^2 )dx.

2 Find:

a

Z

(2e¡x+3)dx b

Z μ

p

x¡

1

p

x

¶

dx c

Z ¡

3+e^2 x¡^1

¢ 2

dx

3 Given that f^0 (x)=x^2 ¡ 3 x+2and f(1) = 3, find f(x).

4 Find the exact value of

Z 3

2

1

p

3 x¡ 4

dx.

5 By differentiating (3x^2 +x)^3 , find

Z

(3x^2 +x)^2 (6x+1)dx.

6 If

Z 4

1

f(x)dx=3, determine:

a

Z 4

1

(f(x)+1)dx b

Z 2

1

f(x)dx¡

Z 2

4

f(x)dx

7 Given that f^00 (x) = 2 sin(2x), f^0 (¼ 2 )=0, and f(0) = 3, find the exact value of f(¼ 2 ).

8 Find

d

dx

(e¡^2 xsinx) and hence find

Z¼

2

0

£

e¡^2 x(cosx¡2 sinx)

¤

dx

4037 Cambridge

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Y:\HAESE\CAM4037\CamAdd_15\435CamAdd_15.cdr Monday, 7 April 2014 4:00:17 PM BRIAN