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Integration Overview

Integration is a bit of an art - some expressions are easy to do and some almost identical expressions have no integral amongst the functions that you know.

Basic integrations

xn ⇒ xn+1/(n+1) n≠-1
x-1 ⇒ ln x
ex ⇒ ex
cos x ⇒ sin x
sin x ⇒ -cos x
Substitution is the inverse of the chain rule. You are looking for a u such that the integrand is a basic function of u times du/dx. Often they will give you the u to use, in which case it is mechanical.

Otherwise you have to look for an obvious candidate.

To lay out your answer

  • Identify your u (or copy it if it is given).
  • Work out du/dx and hence work out dx in terms of du.
  • Change the integrand into a function of u, substitute dx
  • If there are any x's left, you will also have to express them in terms of u

If it is a definite integral, you also have to change the limits of integration from x values into u values.

Hopefully this integral looks like the basic forms above.

There are some cases where the substitution is not so obvious, but in this case it should be given to you.

One example is where there are functions like √(1-x2) in your integrand. In this case a substitution of x=cos u can work really well. There is an example of this here.

Integration by parts is the inverse of the product rule for differentiation.

You need to have at least two things multiplied together in your integrand. Pick these as your u and dv/dx. The one you pick as dv/dx must of course be something you can integrate.

Lay it out clearly and it will be easy to do:

udu/dx
dv/dxv

Then you can write the integral down - (uv) minus the integral of the second column of the table above.

uv - ∫ v du/dx dx

If the integral is definite, don't forget the integration limits on the [uv] term.

Again this only works if the second form is easier to integrate. There is a nice case where what you end up with is the same as you started with: see here.