Change of variables

Definition

Change of variables is a method used to simplify integrals by substituting a new variable for an existing one. This technique often makes the integral easier to evaluate.

5 Must Know Facts For Your Next Test

1. The substitution $u = g(x)$ is commonly used, where $g(x)$ is a differentiable function.
2. After substitution, the differential $dx$ must be expressed in terms of the new variable: $du = g'(x) dx$.
3. The limits of integration must also be changed when dealing with definite integrals.
4. This method can help in evaluating integrals involving composite functions.
5. It's crucial to revert back to the original variable if solving an indefinite integral.

Review Questions

• What is the purpose of using change of variables in integration?
• How do you transform the differential $dx$ when using a substitution $u = g(x)$?
• Why is it important to adjust the limits of integration when dealing with definite integrals?

Related terms

Indefinite Integral: An integral without specified limits, representing a family of antiderivatives.

Definite Integral: An integral with specified upper and lower limits, representing the net area under a curve.

$u$-Substitution: $u$-Substitution is a specific type of change of variables where $u = g(x)$ transforms an integral into one that is easier to evaluate.