Selecting Techniques for Integration (College Board AP® Calculus BC): Revision Note

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. $\frac{d}{dx}\left(\sin x\right)=\cos x$

      • therefore $\int \cos xdx=\sin x+C$

  • See the 'Derivatives & Antiderivatives' study guide

• More complicated integrals can be solved by

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

    • $\int \left(pf\left(x\right)\pm qg\left(x\right)\right)dx=p\int f\left(x\right)dx\pm q\int g\left(x\right)dx$

  • simplifying functions to make them easier to integrate

    • E.g. ${\left(x^2+2\right)}^2=x^4+4x^2+4$

    • or $\frac{5x^3-3}{x^2}=5x-3x^{-2}$

  • 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. $\frac{d}{dx}\left(\sin \left(x^2-5x+4\right)\right)=\left(2x-5\right)\cos \left(x^2-5x+4\right)$, by the chain rule

    • therefore $\int \left(2x-5\right)\cos \left(x^2-5x+4\right)dx=\sin \left(x^2-5x+4\right)+C$

  • 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 $u=x-4$

    • it can be shown that $\int x\sqrt{x-4}dx=\frac{2}{5}{\left(x-4\right)}^{\frac{5}{2}}+\frac{8}{3}{\left(x-4\right)}^{\frac{3}{2}}+C$

  • 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

    • $\int \frac{f\text{'}\left(x\right)}{f\left(x\right)}dx=\ln \left|f\left(x\right)\right|+C$

  • 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

    • $\int \frac{1}{\sqrt{1-x^2}}dx=\arcsin x+C,-1<x<1$

    • or $\int \frac{1}{1+x^2}dx=\mathrm{arc}\tan x+C$

  • E.g. $\int \frac{1}{x^2-6x+13}dx$ can be integrated

    • by first completing the square on the denominator to get $\frac{1}{x^2-6x+13}=\frac{1}{4+{\left(x-3\right)}^2}=\frac{1}{4}·\frac{1}{1+{\left({\displaystyle \frac{x-3}{2}}\right)}^2}$

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

• Some integrals can be simplified by using polynomial long division

  • E.g. $\frac{x^3+6x^2-9x-11}{x-2}$ can be written as $x^2+8x+7+\frac{3}{x-2}$

    • 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

  • $\int u·\frac{dv}{dx}dx=uv-\int \frac{du}{dx}·vdx$

  • e.g. $\int x\cos xdx=x\sin x-\int \sin xdx$

    • See the 'Integration by Parts' study guide

• The value of a definite integral can be found

  • by using the result ${\int }_a^{b}f\left(x\right)dx=F\left(b\right)-F\left(a\right)$, where $F$ is an antiderivative of $f$

    • 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