Integration by Substitution (DP IB Analysis & Approaches (AA): SL): Revision Note

Substitution: reverse chain rule

What is integration by substitution?

• When reverse chain rule is difficult to spot or awkward to use then integration by substitution can be used instead

  • Substitution simplifies the integral by defining an alternative variable (usually$u$) in terms of the original variable (usually$x$)

  • Everything (including “$\mathrm{d}x$” and limits for definite integrals) is then substituted which makes the integration much easier

How do I integrate using substitution?

• STEP 1
  Identify the substitution to be used – it will be the secondary function in the composite function

  • So if you are integrating something like$f\left(g\right(x\left)\right)$, use the substitution $u=g\left(x\right)$

  • E.g. $\int 2x\cos \left(x^2\right)\mathrm{d}x$

    • Use the substitution $u=x^2$

• STEP 2
  Differentiate the substitution and rearrange

  $\frac{\mathrm{d}u}{\mathrm{d}x}$can be treated like a fraction (i.e. “multiply by$\mathrm{d}x$” to get rid of fractions)

  • E.g. $u=x^2\Rightarrow \frac{\mathrm{d}u}{\mathrm{d}x}=2x\Rightarrow \mathrm{d}u=2x\mathrm{d}x$

• STEP 3

  Replace all parts of the integral

  All$x$ terms should be replaced with equivalent$u$ terms, including$\mathrm{d}x$

  (If finding a definite integral change the limits from$x$-values to$u$-values too)

  • E.g. $\int 2x\cos \left(x^2\right)\mathrm{d}x=\int \cos \left(x^2\right)\left(2x\mathrm{d}x\right)=\int \cos u\mathrm{d}u$

• STEP 4

  Integrate and either substitute$x$ back in (indefinite integral) or evaluate using the$u$ limits (definite integral)

  • E.g. $\int \cos u\mathrm{d}u=\sin u+c=\sin \left(x^2\right)+c$

• STEP 5

  Find$c$, the constant of integration, if needed
   

• For definite integrals, a GDC should be able to process the integral without the need for a substitution

  • Be clear about whether or not working is required in a question