Cambridge Additional Mathematics

Integration (Chapter 15) 431

8 Suppose f^0 (x)=psin

¡ 1

2 x

¢

wherepis a constant. f(0) = 1 and f(2¼)=0. Findpand

hence f(x).

9 Consider a functiongsuch that g^00 (x)=¡sin 2x.

Show that the gradients of the tangents to y=g(x) when x=¼ and x=¡¼ are equal.

10 Find f(x) given f^0 (x)=2e¡^2 x and f(0) = 3.

11 A curve has gradient function

p

x+^12 e¡^4 x and passes through (1,0). Find the equation of the

function.

Earlier we saw thefundamental theorem of calculus:

If F(x) is the antiderivative of f(x) where f(x) is continuous on the interval a 6 x 6 b, then the

definite integralof f(x) on this interval is

Zb

a

f(x)dx=F(b)¡F(a).

Zb

a

f(x)dx reads “the integral from x=a to x=b of f(x) with respect tox”

or “the integral from a to b of f(x) with respect tox”.

It is called adefiniteintegral because there are lower and upper limits for the integration, and it therefore

results in a numerical answer.

When calculating definite integrals we can omit the constant of integrationcas this will always cancel out

in the subtraction process.

It is common to write F(b)¡F(a) as [F(x)]ba, and so

Zb

a

f(x)dx=[F(x)]ba=F(b)¡F(a)

Earlier in the chapter we proved the following properties of definite integrals using the fundamental theorem

of calculus:

²

Zb

a

f(x)dx=¡

Za

b

f(x)dx

²

Zb

a

cf(x)dx=c

Zb

a

f(x)dx, cis any constant

²

Zb

a

f(x)dx+

Zc

b

f(x)dx=

Zc

a

f(x)dx

²

Zb

a

[f(x)+g(x)]dx=

Zb

a

f(x)dx+

Zb

a

g(x)dx

G Definite integrals

4037 Cambridge

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Y:\HAESE\CAM4037\CamAdd_15\431CamAdd_15.cdr Monday, 7 April 2014 3:59:48 PM BRIAN