In calculus, the substitution rule is a tool for finding antiderivatives and integrals. Using the fundamental theorem of calculus often requires finding an antiderivative. For this and other reasons, the substitution rule is a relatively important tool for mathematicians. It is the counterpart to the chain rule of differentiation. Calculus is a central branch of mathematics, developed from algebra and geometry, and built on two major complementary ideas. ... In calculus, an antiderivative or primitive function of a given real valued function f is a function F whose derivative is equal to f, i. ... In calculus, the integral of a function is a generalization of area, mass, volume, sum, and total. ... The fundamental theorem of calculus is the statement that the two central operations of calculus, differentiation and integration, are inverses of each other. ... In calculus, the chain rule is a formula for the derivative of the composition of two functions. ... In mathematics, the derivative is one of the two central concepts of calculus. ...

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. In mathematics, a function is a relation, such that each element of a set (the domain) is associated with a unique element of another (possibly the same) set (the codomain, not to be confused with the range). ... In mathematics, interval is a concept relating to the sequence and set-membership of one or more numbers. ... In mathematics, the image of an element x in a set X under the function f : X → Y, denoted by f(x), is the unique y in Y that is associated with x. ... In mathematics, the domain of a function is the set of all input values to the function. ...

Suppose the derivative φ'(t) is integrable on [a,b] and

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.

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 or double angle formula followed by one more substitution. In calculus, and more generally in mathematical analysis, integration by parts is a rule that transforms the integral of products of functions into other, possibly simpler, integrals. ...

Antiderivatives

The substitution rule can be used to determine antiderivatives. One chooses a relation between x and t, determines the corresponding relation between dx and dt by differentiating, and performs the substitutions. An antiderivative for the substituted function can hopefully be determined; the original substitution between x and t is then undone. In calculus, an antiderivative or primitive function of a given real valued function f is a function F whose derivative is equal to f, i. ...

Similar to our first example above, we can determine the following antiderivative with this method:

where C is an arbitrary constant of integration.

Note that there were no integral boundaries to transform, but in the last step we had to revert the original substitution x = t² + 1.

Substitution rule for multiple variables

One may also use substitution when integrating functions of several variables. Here the substitution function (v₁,...,v_n) = φ(u₁,...,u_n ) needs to be one-to-one and continuously differentiable, and the differentials transform as In mathematics, an injective function (or one-to-one function or injection) is a function which maps distinct input values to distinct output values. ...

where det(Dφ)(u₁,...,u_n ) denotes the determinant of the Jacobian matrix containing the partial derivatives of φ . This formula expresses the fact that the absolute value of the determinant of given vectors equals the volume of the spanned parallelepiped. In linear algebra, a determinant is a function depending on n that associates a scalar det(A) to every n×n square matrix A. The fundamental geometric meaning of a determinant is as the scale factor for volume when A is regarded as a linear transformation. ... In vector calculus, the Jacobian is shorthand for either the Jacobian matrix or its determinant, the Jacobian determinant. ... In mathematics, a partial derivative of a function of several variables is its derivative with respect to one of those variables with the others held constant. ... In mathematics, the absolute value (or modulus1) of a real number is its numerical value without regard to its sign. ... In geometry, a parallelepiped or parallelopipedon is a three-dimensional figure like a cube, except that its faces are not squares but parallelograms. ...

More precisely, the change of variables formula is stated in the following theorem:

Theorem. Let U, V  be open sets in Rⁿ and φ : U → V  a bijective differentiable function with continuous partial derivatives. Then for any integrable real-valued function f  on V  In mathematics, a bijection, bijective function, or one-to-one correspondence is a function that is both injective (one-to-one) and surjective (onto), and therefore bijections are also called one-to-one and onto. ...