Editing Integration by substitution (section)

=== Statement for definite integrals ===
Let <math>g:[a,b]\to I</math> be a [[differentiable function]] with a [[continuous function|continuous]] derivative, where <math>I \subset \mathbb{R}</math> is an [[interval (mathematics)|interval]]. Suppose that <math>f:I\to\mathbb{R}</math> is a [[continuous function]]. Then:<ref>{{harvnb|Briggs|Cochran|2011|p=361}}</ref>
<math display="block">\int_a^b f(g(x))\cdot g'(x)\, dx = \int_{g(a)}^{g(b)} f(u)\ du. </math>

In Leibniz notation, the substitution <math>u=g(x)</math> yields: 
<math display="block">\frac{du}{dx} = g'(x).</math>
Working heuristically with [[infinitesimal]]s yields the equation
<math display="block">du = g'(x)\,dx,</math>
which suggests the substitution formula above.  (This equation may be put on a rigorous foundation by interpreting it as a statement about [[differential form]]s.)  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 [[function composition|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 of a function|differential]] with the differential of the trigonometric function.