Integration by substitution

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.

Before stating the result rigorously, consider a simple case using indefinite integrals.

Compute ${\textstyle \int (2x^{3}+1)^{7}(x^{2})\,dx.}$

Set ${\displaystyle u=2x^{3}+1.}$ This means ${\textstyle {\frac {du}{dx}}=6x^{2},}$ or as a differential form, ${\textstyle du=6x^{2}\,dx.}$ Now: $${\displaystyle {\begin{aligned}\int (2x^{3}+1)^{7}(x^{2})\,dx&={\frac {1}{6}}\int \underbrace {(2x^{3}+1)^{7}} _{u^{7}}\underbrace {(6x^{2})\,dx} _{du}\\&={\frac {1}{6}}\int u^{7}\,du\\&={\frac {1}{6}}\left({\frac {1}{8}}u^{8}\right)+C\\&={\frac {1}{48}}(2x^{3}+1)^{8}+C,\end{aligned}}}$$ where ${\displaystyle C}$ 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. $${\displaystyle {\frac {d}{dx}}\left[{\frac {1}{48}}(2x^{3}+1)^{8}+C\right]={\frac {1}{6}}(2x^{3}+1)^{7}(6x^{2})=(2x^{3}+1)^{7}(x^{2}).}$$ For definite integrals, the limits of integration must also be adjusted, but the procedure is mostly the same.

Let ${\displaystyle g:[a,b]\to I}$ be a differentiable function with a continuous derivative, where ${\displaystyle I\subset \mathbb {R} }$ is an interval. Suppose that ${\displaystyle f:I\to \mathbb {R} }$ is a continuous function. Then: $${\displaystyle \int _{a}^{b}f(g(x))\cdot g'(x)\,dx=\int _{g(a)}^{g(b)}f(u)\ du.}$$

In Leibniz notation, the substitution ${\displaystyle u=g(x)}$ yields: $${\displaystyle {\frac {du}{dx}}=g'(x).}$$ Working heuristically with infinitesimals yields the equation $${\displaystyle du=g'(x)\,dx,}$$ which suggests the substitution formula above. (This equation may be put on a rigorous foundation by interpreting it as a statement about differential forms.) One may view the method of integration by substitution as a partial justification of Leibniz's notation for integrals and derivatives.

The formula is used to transform one integral into another integral that is easier to compute. Thus, the formula can be read from left to right or from right to left in order to simplify a given integral. When used in the former manner, it is sometimes known as u-substitution or w-substitution in which a new variable is defined to be a function of the original variable found inside the composite function multiplied by the derivative of the inner function. The latter manner is commonly used in trigonometric substitution, replacing the original variable with a trigonometric function of a new variable and the original differential with the differential of the trigonometric function.