Partial integration
To differentiate a product:
By integrating this:
For 
To get the idea, compare differentiation: .
For 
To calculate, complete the square and substitute, for example:
Substitute .
The goal is to rewrite the integral by substitution in order for it to take one of these forms:
For 
To calculate, complete the square and substitute (using a linear substitution) in order for it to take one of these forms:
For 
For 
with .
Calculate J by substituting .
The equation has no real solutions.
Calculate K by substituting to arrive at:
If , then:
For , recurse.
Let R(x) be a rational function.
For 
Substitute to get a rational function in t.
For 
Substitute to get a rational function in t.
For 
Substitute to get a rational function in t.
For 
Substitute to get an exponential integral.
For 
Substitute to get an exponential integral.
For 
Substitute or
to get a function with just sin and cos in them.
For 
complete the square.
For
with a polynom P
partial integration to decrease the order of the polynom.
For
with a polynom P
partial integration to decrease the order of the polynom.
For
with a polynom P
partial integration to decrease the order of the polynom.
Uneigentliche Integrale
Locally Integrable
A function is locally integrable if it is at least integrable in a range [a,x).
One can also do this with the range (x,a].
Limit
By crossing the border, one can define the Uneigentliches Integral like this:
Multi-dimensional partial integration
Let Ω be a subset of R^n with a piecemeal smooth border ∂Ω. Let the orientation of the border be the normal vector n.
Let v be a differentiable vector field on the environment Ω and φ a steady differentiable scalar field on Ω.
Let the abbreviation dS mean n⋅dS.
Then follows, reminiscent of the product integral rule: