# Integration Techniques

## Integration by Substitution

## Video

This video from *Khan Academy* works through an example of integration by substitution, and discusses how to recognise when to use it, and how to choose an appropriate $u$.

Note: All *Khan Academy* content is available for free at www.khanacademy.org.

This video from *Khan Academy* gives an example of computing definite integrals that require substitution. It covers two different methods of evaluating the integral at the limits.

Note: All *Khan Academy* content is available for free at www.khanacademy.org.

This video from *Khan Academy* discusses an example of integration by substitution where the differential factor $du$ does not exactly equal the remainder of the integrand after $u$ has been chosen.

Note: All *Khan Academy* content is available for free at www.khanacademy.org.

## Notes

These notes from MathCentre describe the integration by substitution technique and how to use it.

A pdf of this document can be downloaded here.

## Interactive Exercises

You can try different integration by substitution problems in these NUMBAS exercises.

Please click here to view this NUMBAS question in another window.

Please click here to view this NUMBAS question in another window.

Please click here to view this NUMBAS question in another window.

## Integration Using Trigonometric Substitution

## Video

This video from *Khan Academy* gives an example of computing an integral using a trigonometric substitution. It also explains the reasoning behind why the substitution is made.

Note: All *Khan Academy* content is available for free at www.khanacademy.org.

## Notes

Page 5 onwards of these notes from MathCentre show how to use trigonometric substitution to evaluate certain kinds of integrals. There are exercises at the end with answers given for you to practise with.

A pdf of this document can be downloaded here.

## Interactive Exercises

This NUMBAS exercise lets you practice computing some integrals that require trigonometric substitutions.

Please click here to view this NUMBAS question in another window.

## Integration by Partial Fractions

## Videos

This video from *Krista King* shows how to compute an integral involving an algebraic fraction with a repeated linear factor in the denominator by using partial fractions.

Note: All *Krista King* content is available for free at www.youtube.com/c/Integralcalc.

Krista King’s Website: www.kristakingmath.com .

## Notes

These notes from MathCentre show how to compute integrals of rational functions, in particular by using partial fractions to simplify the integration process.

A pdf of this document can be downloaded here.

## Integration by Parts

## Videos

This video from *patrickJMT* shows how to use integration by parts to compute the integral of a product of functions.

Note: All *patrickJMT* content is available for free at www.youtube.com/user/patrickJMT.

Patrick JMT’s Website: www.patrickjmt.com.

This video from *Khan Academy* shows how to use integration by parts to evaluate a definite integral.

Note: All *Khan Academy* content is available for free at www.khanacademy.org.

This video from *patrickJMT* discusses a variation of an integration by parts question where two rounds of integration are required, and the end result is obtained by using an algebraic trick.

Note: All *patrickJMT* content is available for free at www.youtube.com/user/patrickJMT.

Patrick JMT’s Website: www.patrickjmt.com.

## Notes

These notes from MathCentre give examples of both definite and indefinite integration by parts questions.

A pdf of this document can be downloaded here.

## Interactive Exercises

You can practise different integration by parts exercises with these NUMBAS exercises.

Please click here to view this NUMBAS question in another window.

Please click here to view this NUMBAS question in another window.

Please click here to view this NUMBAS question in another window.