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  • A-Level Further Maths

Further Integration

This is a bit of an odd notes sheet - its just an amalgamation of all that extra integration stuff:

  • improper integrals

  • the mean value of a function

  • inverse trig

  • partial fractions

  • volumes of revolution


Improper Integrals

In single maths, you learn how to integrate to find the area enclosed between a curve and the x-axis between two fixed limits. An improper integral is when one or both limits are infinite, or when the function is undefined at a certain point in the given interval.

Improper integrals, further integration. Free online a-level further maths pure notes. EngineeringNotes.net, EngineeringNotes, Engineering Notes.
An improper integral does not always exist: if it does, it is convergent; if it does not, it is divergent.

One Infinite/Undefined Limit

To see if an improper integral with one infinite and one finite limit is convergent you need to substitute the infinite limit for t, and consider what happens as the function tends towards infinity:

Improper integrals, further integration. Free online a-level further maths pure notes. EngineeringNotes.net, EngineeringNotes, Engineering Notes.
If the integral part with t in it did not tend to 0, but instead tended to ∞, the function would be divergent.

Two Infinite/Undefined Limits

When both limits are infinite or undefined, you need to separate the integral into two improper integrals each with one infinite and one finite limit.

The finite limit of both parts must be the same.

It is generally good to use a simple value such as x=0, as this makes workings a lot simpler.

Improper integrals, further integration. Free online a-level further maths pure notes. EngineeringNotes.net, EngineeringNotes, Engineering Notes.

Even if only one of the two separated integrals does not converge, the whole integral will not converge to a defined value.