In calculus, integration by substitution, also known as u-substitution, reverse chain rule or change of variables, is a method for evaluating integrals and antiderivatives. It is the counterpart to the chain rule for differentiation, and can loosely be thought of as using the chain rule "backwards." This involves differential forms.
Substitution for a single variable
Introduction (indefinite integrals) Before stating the result rigorously, consider a simple case using indefinite integrals. Compute ∫ ( 2 x 3 + 1 ) 7 ( x 2 ) d x . {\textstyle \int (2x^{3}+1)^{7}(x^{2})\,dx.}
Set u = 2 x 3 + 1. {\displaystyle u=2x^{3}+1.} This means d u d x = 6 x 2 , {\textstyle {\frac {du}{dx}}=6x^{2},} or as a differential form, d u = 6 x 2 d x . {\textstyle du=6x^{2}\,dx.} Now consider ∫ u 7 d u . {\textstyle \int u^{7}du.} :
∫ ( 2 x 3 + 1 ) 7 ( x 2 ) d x = 1 6 ∫ ( 2 x 3 + 1 ) 7 ⏟ u 7 ( 6 x 2 ) d x ⏟ d u = 1 6 ∫ u 7 d u = 1 6 ( 1 8 u 8 ) + C = 1 48 ( 2 x 3 + 1 ) 8 + C , {\displaystyle {\begin{aligned}\int (2x^{3}+1)^{7}(x^{2})\,dx\\={\frac {1}{6}}\int \underbrace {(2x^{3}+1)^{7}} _{u^{7}}\underbrace {(6x^{2})\,dx} _{du}\\={\frac {1}{6}}\int u^{7}\,du={\frac {1}{6}}\left({\frac {1}{8}}u^{8}\right)+C\\={\frac {1}{48}}(2x^{3}+1)^{8}+C,\end{aligned}}}
where C {\displaystyle C} is an arbitrary constant of integration. This procedure is frequently used, but not all integrals are of a form that permits its use. In any event, the result should be verified by differentiating and comparing to the original integrand.
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