Integration by substitution

In calculus, integration by substitution, also known as u-substitution, is a method for finding integrals. Using the fundamental theorem of calculus often requires finding an antiderivative. For this and other reasons, integration by substitution is an important tool in mathematics. It is the counterpart to the chain rule of differentiation.

Let $I \subseteq ℝ$ be an interval and $φ : [a, b] \to I$ be a differentiable function with integrable derivative. Suppose that $ƒ : I \to ℝ$ is a continuous function. Then

In other notation: the substitution $x = φ (t)$ yields

and thus, formally, $dx = φ'(t) dt$ , which is the required substitution for $dx$ . (One could view the method of integration by substitution as a 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 used from left to right or from right to left in order to simplify a given integral. When used in the latter manner, it is sometimes known as u-substitution or w-substitution.

Integration by substitution can be derived from the fundamental theorem of calculus as follows. Let $ƒ$ and $φ$ be two functions satisfying the above hypothesis that $ƒ$ is continuous on $I$ and $φ'$ is integrable on the closed interval $[a, b]$ . Then the function $ƒ (φ (t)) φ'(t)$ is also integrable on $[a, b]$ . Hence the integrals

and

in fact exist, and it remains to show that they are equal.

Since $ƒ$ is continuous, it has an antiderivative $F$ . The composite function $F \circ φ$ is then defined. Since $F$ and $φ$ are differentiable, the chain rule gives

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