解釋integration by substitution

∫f(x) dx
If we can find u (in terms of x) such that we can express this in the form
∫g(u) (du/dx) dx,
∫f(x) dx = ∫g(u) (du/dx) dx = ∫g(u) du

Example:
∫cos 2x dx
Let u=2x du/dx=2
=(1/2)∫2 cos 2x dx
=(1/2)∫cos u du
=(sin u)/2 + C
Remember to substitute value of u into the answer!
=(sin 2x)/2 + C

We can reverse this rule:
We can let x = v (in terms of u), we have
∫f(x) dx = ∫f(v)v' du

Example:
∫sqrt(1-x^2) dx
Let x=sin u dx/du=cos u
=∫sqrt(1-sin^2 u) cos u du
=∫cos^2 u du
By using double angle formula one more substitution,
=(sin 2u)/2 + u/2 + C
x=sin u, u=arcsin x
=(sin (2arcsin x))/2 + (arcsin x)/2 + C

We can use this rule for definite intergration:
∫(from a to b) f((g(x))g'(x) dx = ∫(from g(a) to g(b)) f(x) dx
Use of this rule is similar to above, except we have to change the limits and we DON'T have to substitute the value back.

http://www.math.ncu.edu.tw/~yu/ecocal95/boards/lec37_c_95.pdf