5.5 Substitution  (Page 3/6)

Then

and we have the desired result.

Substitution may be only one of the techniques needed to evaluate a definite integral. All of the properties and rules of integration apply independently, and trigonometric functions may need to be rewritten using a trigonometric identity before we can apply substitution. Also, we have the option of replacing the original expression for u after we find the antiderivative, which means that we do not have to change the limits of integration. These two approaches are shown in [link] .

Key concepts

• Substitution is a technique that simplifies the integration of functions that are the result of a chain-rule derivative. The term ‘substitution’ refers to changing variables or substituting the variable u and du for appropriate expressions in the integrand.
• When using substitution for a definite integral, we also have to change the limits of integration.

Key equations

• Substitution with Indefinite Integrals
  ${\displaystyle \int f\left[g\left(x\right)\right]g^{\prime }\text{(}x)dx={\displaystyle \int f\left(u\right)du}}=F\left(u\right)+C=F\left(g\left(x\right)\right)+C$
• Substitution with Definite Integrals
  ${\displaystyle {\int }_a^{b}f\left(g\left(x\right)\right)g\text{'}\left(x\right)dx={\displaystyle {\int }_{g\left(a\right)}^{g\left(b\right)}f\left(u\right)du}}$

In the following exercises, verify each identity using differentiation. Then, using the indicated u -substitution, identify f such that the integral takes the form ${\displaystyle \int f\left(u\right)du}.$