Cambridge Additional Mathematics

424 Integration (Chapter 15)

We can check that an integral

is correct by differentiating the

answer. It should give us the

, the function we

originally integrated.

integrand

7 If y= cos 2x, find

dy

dx

. Hence find

Z

sin 2xdx.

8 If y= sin(1¡ 5 x), find dy

dx

. Hence find

Z

cos(1¡ 5 x)dx.

9 By considering

d

dx

(x^2 ¡x)^3 , find

Z

(2x¡1)(x^2 ¡x)^2 dx.

10 Prove the rule

Z

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

Z

f(x)dx+

Z

g(x)dx.

11 Find

dy

dx

if y=

p

1 ¡ 4 x. Hence find

Z

1

p

1 ¡ 4 x

dx.

InChapter 13we developed a set of rules to help us differentiate functions more efficiently:

Function Derivative Name

c, a constant 0

mx+c, mandcare constants m

xn nxn¡^1 power rule

cu(x) cu^0 (x)

u(x)+v(x) u^0 (x)+v^0 (x) addition rule

u(x)v(x) u^0 (x)v(x)+u(x)v^0 (x) product rule

u(x)

v(x)

u^0 (x)v(x)¡u(x)v^0 (x)

[v(x)]^2

quotient rule

y=f(u) where u=u(x)

dy

dx

=

dy

du

du

dx

chain rule

ex ex

ef(x) ef(x)f^0 (x)

lnx

1

x

F Integrating

(^0) (x)
f(x)
[f(x)]n n[f(x)]n¡^1 f^0 (x)
sinx cosx
cosx ¡sinx
tanx sec^2 x

E Rules for integration

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Additional Mathematics
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