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 $\frac{2}{3} u^{3/2} + C = \frac{2}{3} (1 + x^2)^{3/2} + C$.