| ∫ uw dx = u ∫ w dx − ∫ | du | { ∫ w dx } dx. |
| dx |
This is the rule of “integration by parts.”
As an example we have
| ∫ xeax dx = x | eax | − ∫ | eax | dx = ( | x | − | 1 | ) eax. |
| a | a | a | a2 |
When we introduce a new variable z in place of x, by means of an equation giving x in terms of z, we express ƒ(x) in terms of z. Let φ(z) denote the function of z into which ƒ(x) is transformed. Then from the equation
| dx = | dx | dz |
| dz |
we deduce the equation
| ∫ ƒ(x) dx = ∫ φ(z) | dx | dz. |
| dz |
As an example, in the integral