Table 4.4.1 lists the rule for making a substitution, or change of variable, in an integral.
Substitution Rule for Integrals
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1.
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and satisfy suitable conditions
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⇒
1.
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2.
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Table 4.4.1 Substitution in definite and indefinite integrals
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If is the antiderivative obtained when the substitution is made in the indefinite integral, then is an indefinite integral in terms of the original variable, ; correspondingly, the value of the definite integral is given by = .
Alternate forms for the substitution are and . In either of these cases, obtain by solving explicitly for and differentiating, or by differentiating implicitly. No matter how the substitution rule is given, an explicit solution for is needed for expressing the indefinite integral in terms of .