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Objectives

  • Use substitution to evaluate indefinite integrals
  • Use substitution to evaluate definite integrals

Summary

To find algebraic formulas for antiderivatives of more complicated algebraic functions, we need to think carefully about how we can reverse known differentiation rules. To that end, it is essential that we understand and recall known derivatives of basic functions, as well as the standard derivative rules.

The indefinite integral provides notation for antiderivatives. When we write " ∫ f ( x ) d x ", we mean "the general antiderivative of f ". In particular, if we have functions f and F such that F ′ = f , the following two statements say the exact thing:

d d x [ F ( x ) ] = f ( x )   and ∫ f ( x ) d x = F ( x ) + C

That is, f is the derivative of F , and F is an antiderivative of f .

The technique of u-substitution helps us evaluate indefinite integrals of the form ∫ f ( g ( x ) ) g ′ ( x ) d x through substitutions u = g ( x ) and d u = g ′ ( x ) d x so that

∫ f ( g ( x ) ) g ′ ( x ) d x = ∫ f ( u ) d u

A key part of choosing the expression in x to be represented by u is the identification of a function-derivative pair. To do so, we often look for an "inner" function g ( x ) that is part of a composite function, while investigating whether g ′ ( x ) (or a constant multiple of g ′ ( x ) ) is present as a multiplying factor of an integrand.

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