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 an important branch of mathematics. ... 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 an extension of the concept of a sum. ... The fundamental theorem of calculus is the statement that the two central operations of calculus, differentiation and integration, are inverse functions of one another. ... In calculus, the chain rule is a formula for the derivative of the composite of two functions. ... In mathematics, a derivative is the rate of change of a quantity (e. ...

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 (also known as range) is contained in the domain of f. Partial plot of a function f. ... In mathematics, a smooth function is one that is infinitely differentiable, i. ... 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 range of a function is the set of all output values produced by that function. ... 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's notation: 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 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.

Examples

Consider the integral

By using the substitution t = x² + 1, we obtain dt = 2x dx and

Here we used the substitution rule from right to left. Note how the lower limit x = 0 was transformed into t = 0² + 1 = 1 and the upper limit x = 2 into t = 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 a 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 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 (as opposed to the total derivative, in which all variables are allowed to vary). ... In mathematics, the absolute value (or modulus1) of a real number is its numerical value without regard to its sign. ... In geometry, a parallelepiped (pronounced ; meaning of parallel planes) 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  an injective differentiable function with continuous partial derivatives, the Jacobian of which is nonzero for every x in U. Then for any real-valued, compactly supported, continuous function f, with support connected in φ(U), An injective function. ...