Cambridge Additional Mathematics

428 Integration (Chapter 15)

If we are given the second derivative we need to integrate twice to find the function. This creates two

integrating constants, so we need two other facts about the curve in order to determine these constants.

Example 10 Self Tutor

Find f(x) given that f^00 (x)=12x^2 ¡ 4 , f^0 (0) =¡ 1 , and f(1) = 4.

If f^00 (x)=12x^2 ¡ 4

then f^0 (x)=

12 x^3

3

¡ 4 x+c fintegrating with respect toxg

) f^0 (x)=4x^3 ¡ 4 x+c

But f^0 (0) =¡ 1 ,soc=¡ 1

Thus f^0 (x)=4x^3 ¡ 4 x¡ 1

) f(x)=

4 x^4

4

¡

4 x^2

2

¡x+d fintegrating againg

) f(x)=x^4 ¡ 2 x^2 ¡x+d

But f(1) = 4,so 1 ¡ 2 ¡1+d=4 and hence d=6

Thus f(x)=x^4 ¡ 2 x^2 ¡x+6

EXERCISE 15E.2

1 Find f(x) given that:

a f^0 (x)=2x¡ 1 and f(0) = 3 b f^0 (x)=3x^2 +2x and f(2) = 5

c f^0 (x)=ex+

1

p

x

and f(1) = 1 d f^0 (x)=x¡

2

p

x

and f(1) = 2

2 Find f(x) given that:

a f^0 (x)=x^2 ¡4 cosx and f(0) = 3 b f^0 (x) = 2 cosx¡3 sinx and f

¡¼

4

¢

=p^12

3 Find f(x) given that:

a f^00 (x)=2x+1, f^0 (1) = 3, and f(2) = 7

b f^00 (x)=15

p

x+

3

p

x

, f^0 (1) = 12, and f(0) = 5

c f^00 (x) = cosx, f^0 (¼ 2 )=0, and f(0) = 3

d f^00 (x)=2x and the points (1,0) and (0,5) lie on the curve.

In this section we deal with integrals of functions which are composite with the linear function ax+b.

Notice that

d

dx

³

1

a

eax+b

́

=

1

a

eax+b£a=eax+b

)

Z

eax+bdx=

1

a

eax+b+c for a 6 =0

F INTEGRATING f(ax+b)

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Additional Mathematics
Y:\HAESE\CAM4037\CamAdd_15\428CamAdd_15.cdr Monday, 7 April 2014 3:59:26 PM BRIAN