Method of Substitution

5.5 The Substitution Rule

• The Substitution Rule is a method used to integrate indefinite integrals, denoted as \int f(x) dx .

The Fundamental Theorem of Calculus

• To evaluate a definite integral \int_a^b f(x) dx , we first need to find its indefinite counterpart, \int f(x) dx .

Method of Substitution

• The method of substitution is a technique to simplify integration by changing the variable.

Example

• Consider the integral \int 2x \sqrt{1 + x^2} dx .
• The differential of 1 + x^2 is d(1 + x^2) = (1 + x^2)' dx = 2x dx .
• Rewrite the integral as \int \sqrt{1 + x^2} \cdot 2x dx = \int \sqrt{1 + x^2} d(1 + x^2) .
• Let u = 1 + x^2 be the substitution, then \int \sqrt{1 + x^2} d(1 + x^2) = \int \sqrt{u} du .
• Integrating \int \sqrt{u} du gives \frac{2}{3} u^{3/2} + C = \frac{2}{3} (1 + x^2)^{3/2} + C .