Selecting Techniques for Integration (College Board AP® Calculus BC): Study Guide

Selecting techniques for integration

How do I choose the correct procedure for integrating?

• You should be familiar with all the different methods for working out indefinite or definite integrals

  • This way you can choose the most appropriate method to use for an exam question

• The simplest way to integrate is to use antiderivatives

  • If you recognise a function being integrated as the derivative of a standard function

    • Then use the fact that differentiation and integration are inverse operations

    • E.g.

      • therefore

  • See the 'Derivatives & Antiderivatives' study guide

• More complicated integrals can be solved by

  • using standard results for sums, differences and constant multiples of integrals

    • 

  • simplifying functions to make them easier to integrate

    • E.g.

    • or

  • See the 'Indefinite Integral Rules' study guide

• Integrals involving composite functions can sometimes be solved by inspection (sometimes known as the 'reverse chain rule')

  • E.g. , by the chain rule

    • therefore

  • See the 'Integrals of Composite Functions' study guide

• Even trickier integrals can sometimes be solved by using u-substitution

  • E.g. by using the substitution

    • it can be shown that

  • u-substitution is also very effective for evaluating definite integrals

  • See the 'Integration Using Substitution' study guide

• There are different methods for integrating rational functions

  • If the top is the derivative of the bottom then you can integrate quickly

    • 

  • If the bottom factorises then you can use partial fractions

• Some integrals can be solved by using completing the square

  • These integrals will usually involve variations of the standard results

    • 

    • or

  • E.g. can be integrated

    • by first completing the square on the denominator to get

  • See the 'Integration Using Completing the Square' study guide

• Some integrals can be simplified by using polynomial long division

  • E.g. can be written as

    • which is much easier to integrate

  • See the 'Integration Using Long Division' study guide

• Some integrals containing a product of two functions can be found using integration by parts

  • 

  • e.g.

    • See the 'Integration by Parts' study guide

• The value of a definite integral can be found

  • by using the result , where is an antiderivative of

    • See the 'Evaluating Definite Integrals' study guide

  • or sometimes by using basic geometric area formulas

    • See the 'Accumulation of Change' study guide

  • or by using the properties of definite integrals

    • See the 'Properties of Definite Integrals' study guide

What else do I need to know about integration?

• Be sure that you are able to work out the value of a constant of integration

  • See the 'Constant of Integration' study guides

• You should be able to approximate the value of a definite integral using Riemann sums and trapezoidal sums

  • See the 'Riemann Sums' and 'Trapezoidal Sums' study guides

• Finally, be sure that you are familiar with the background theory of integration

  • For example

    • Accumulation of change and accumulation functions

    • Definite integrals as a limit of Riemann sums

    • The fundamental theorem of calculus

  • You may need to recognize the ideas and notation from these areas to answer exam questions on integration