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Home > Substitution rule

In calculus, the substitution rule is an important tool for finding antiderivatives and integrals. It is the counterpart to the chain rule of differentiation.

Suppose f(x) is an integrable function, and φ(t) is a continuously differentiable function which is defined on the interval [a, b] and whose image is contained in the domain of f. Then

The formula is best remembered using Leibniz' formalism: the substitution x = φ(t) yields dx/dt = φ'(t) and thus formally dx = φ'(t) dt, which is precisely the required substitution for dx. (In fact, one may view the substitution rule as a major justification of the Leibniz formalism for integrals and derivatives.)

The formula is used to transform an integral into another one which (hopefully) is easier to determine. 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.

1 Examples

Consider the integral

By using the substitution x = t² + 1, we obtain dx = 2t dt and

Here we used the substitution rule "from right to left". Note how the lower limit t = 0 was transformed into x = 0² + 1 = 1 and the upper limit t = 2 into x = 2² + 1 = 5.

For the integral

the formula needs to be used from left to right: the substitution x = sin(t), dx = cos(t) dt is useful, because √(1-sin²(t)) = cos(t):

The resulting integral can be computed using integration by parts.

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Topics: Substitution Rule Function Formula Integral Left Transform...

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