Integration by substitution and other antidifferentiation techniques

Learning objectives

Recognize an integrand of the form f(g(x)) · g′(x) and choose a substitution u = g(x) that reverses the chain rule, including cases that need a constant multiple adjustment

Evaluate indefinite integrals by substitution and express the result in terms of the original variable

Evaluate definite integrals by substitution, either by changing the limits of integration to values of u or by substituting back before applying the original limits

Integrate rational functions in which the degree of the numerator is at least that of the denominator by first carrying out long division

Integrate expressions by completing the square in the denominator to obtain an arctan or arcsin form

Select an appropriate antidifferentiation technique for a given integrand, including recognizing when algebraic simplification alone is enough