Skills Review for Other Strategies for Integration

Learning Outcome

  • Use substitution to evaluate indefinite integrals

In the Other Strategies for Integration section, we will essentially use a table of integration formulas to evaluate integrals. The most important step when using an integration table is to build the exact integration table formula using the integral we are asked to evaluate. This can sometimes be tricky. Here we will review u-substitution techniques that can be useful when building the desired integration table formula.

Use Substitution to Evaluate Indefinite Integrals

Substitution is where we substitute part of the integrand with the variable [latex]u[/latex] and part of the integrand with du. It is also referred to as change of variables because we are changing variables to obtain an expression that is easier to work with for applying the integration rules.

Substitution with Indefinite Integrals

Let [latex]u=g(x),,[/latex] where [latex]{g}^{\prime }(x)[/latex] is continuous over an interval, let [latex]f(x)[/latex] be continuous over the corresponding range of [latex]g[/latex], and let [latex]F(x)[/latex] be an antiderivative of [latex]f(x).[/latex] Then,

[latex]\begin{array}{cc} {\displaystyle\int f\left[g(x)\right]{g}^{\prime }(x)dx}\hfill & = {\displaystyle\int f(u)du}\hfill \\ & =F(u)+C\hfill \\ & =F(g(x))+C.\hfill \end{array}[/latex]

 

The following steps should be followed when integrating by substitution:

  1. Look carefully at the integrand and select an expression [latex]g(x)[/latex] within the integrand to set equal to [latex]u[/latex]. Let’s select [latex]g(x).[/latex] such that [latex]{g}^{\prime }(x)[/latex] is also part of the integrand.
  2. Substitute [latex]u=g(x)[/latex] and [latex]du={g}^{\prime }(x)dx.[/latex] into the integral.
  3. We should now be able to evaluate the integral with respect to [latex]u[/latex]. If the integral can’t be evaluated we need to go back and select a different expression to use as [latex]u[/latex].
  4. Evaluate the integral in terms of [latex]u[/latex].
  5. Write the result in terms of [latex]x[/latex] and the expression [latex]g(x).[/latex]

Example: Evaluating an Indefinite Integral Using Substitution

Use substitution to find the antiderivative of [latex]\displaystyle\int 6x{(3{x}^{2}+4)}^{4}dx.[/latex]

Example: Evaluating an Indefinite Integral Using Substitution

Use substitution to find the antiderivative of [latex]\displaystyle\int z\sqrt{{z}^{2}-5}dz.[/latex]

Try It

Use substitution to find the antiderivative of [latex]\displaystyle\int {x}^{2}{({x}^{3}+5)}^{9}dx.[/latex]