Ta b l e 9. 1f(x)∫
f(x)dx(arbitrary constant omitted)xα α =− 1 xα+^1 /(α+ 1 )α =− 1
1
x ln|x|
cosx sinx
sinx −cosx
sec^2 x tanx
ex ex
1 /√
a^2 −x^2 sin−^1 (x/a)
a/(a^2 +x^2 ) tan−^1 (x/a)
u′(x)
u(x) ln|u(x)|
u′(x)eu(x) eu(x)- have the standard integrals (at least) at your fingertips
- check your answers by differentiation as often as possible – provides
practice as well as confirmation - there may be more than one way to do an integral – for example
∫
x+ 1
x^2 +x− 6
dxcan be done by partial fractions or by the substitutionu=x^2 +x− 6- there may benowaytodoanintegral–forexample
∫
ex
2
dxsimply cannot be integrated in terms of elementary functions- integration skills build up sequentially – for example:
(i) first learn
∫
dx
x=ln|x|(ii) then∫
dx
x− 1=ln|x− 1 |(iii) then by partial fractions
∫
dx
x^2 − 1=∫ (
1
2 (x− 1 )−1
2 (x− 1 ))
dxSolution to review question 9.1.2
A.These are all particular cases of
∫
xαdx=xα+^1
α+ 1