## 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

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:

where .