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
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Using substitution to evaluate a trigonometric integral
Use substitution to evaluate
Let us first use a trigonometric identity to rewrite the integral. The trig identity
allows us to rewrite the integral as
Then,
We can evaluate the first integral as it is, but we need to make a substitution to evaluate the second integral. Let
Then,
or
Also, when
and when
Expressing the second integral in terms of
u , we have
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
Substitution with Definite Integrals
Why is
u -substitution referred to as
change of variable ?
In the following exercises, verify each identity using differentiation. Then, using the indicated
u -substitution, identify
f such that the integral takes the form
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