Integration by substitution

Technique in integral evaluation 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={