Substitution can be used with definite integrals, too. However, using substitution to evaluate a definite integral requires a change to the limits of integration. If we change variables in the integrand, the limits of integration change as well.
Substitution with Definite Integrals
Let [latex]u=g(x)[/latex] and let [latex]{g}^{\text{′}}[/latex] be continuous over an interval [latex]\left[a,b\right],[/latex] and let [latex]f[/latex] be continuous over the range of [latex]u=g(x).[/latex] Then,
Although we will not formally prove this theorem, we justify it with some calculations here. From the substitution rule for indefinite integrals, if [latex]F(x)[/latex] is an antiderivative of [latex]f(x),[/latex] we have
example: Using Substitution to Evaluate a Definite Integral
Use substitution to evaluate [latex]{\displaystyle\int }_{0}^{1}{x}^{2}{(1+2{x}^{3})}^{5}dx.[/latex]
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Let [latex]u=1+2{x}^{3},[/latex] so [latex]du=6{x}^{2}dx.[/latex] Since the original function includes one factor of [latex]x[/latex]2 and [latex]du=6{x}^{2}dx,[/latex] multiply both sides of the du equation by [latex]1\text{/}6.[/latex] Then,
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Use substitution to evaluate the definite integral [latex]{\displaystyle\int }_{-1}^{0}y{(2{y}^{2}-3)}^{5}dy.[/latex]
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[latex]\frac{91}{3}[/latex]
example: Using Substitution with an Exponential Function
Use substitution to evaluate [latex]{\displaystyle\int }_{0}^{1}x{e}^{4{x}^{2}+3}dx.[/latex]
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Let [latex]u=4{x}^{3}+3.[/latex] Then, [latex]du=8xdx.[/latex] To adjust the limits of integration, we note that when [latex]x=0,u=3,[/latex] and when [latex]x=1,u=7.[/latex] So our substitution gives
Use substitution to evaluate [latex]{\displaystyle\int }_{0}^{1}{x}^{2} \cos \left(\frac{\pi }{2}{x}^{3}\right)dx.[/latex]
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[latex]\frac{2}{3\pi }\approx 0.2122[/latex]
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 [latex]u[/latex] after we find the antiderivative, which means that we do not have to change the limits of integration. These two approaches are shown in the following examples.
example: Using Substitution to Evaluate a Trigonometric Integral
Use substitution to evaluate [latex]{\displaystyle\int }_{0}^{\pi \text{/}2}{ \cos }^{2}\theta d\theta .[/latex]
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Let us first use a trigonometric identity to rewrite the integral. The trig identity [latex]{ \cos }^{2}\theta =\frac{1+ \cos 2\theta }{2}[/latex] allows us to rewrite the integral as
We can evaluate the first integral as it is, but we need to make a substitution to evaluate the second integral. Let [latex]u=2\theta .[/latex] Then, [latex]du=2d\theta ,[/latex] or [latex]\frac{1}{2}du=d\theta .[/latex] Also, when [latex]\theta =0,u=0,[/latex] and when [latex]\theta =\pi \text{/}2,u=\pi .[/latex] Expressing the second integral in terms of [latex]u[/latex], we have
Watch the following video to see the worked solution to Example: Using Substitution to Evaluate a Trigonometric Integral.
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