Method of Substitution

5.5 The Substitution Rule

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

The Fundamental Theorem of Calculus

  • To evaluate a definite integral ∫abf(x)dx\int_a^b f(x) dx, we first need to find its indefinite counterpart, ∫f(x)dx\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 ∫2x1+x2dx\int 2x \sqrt{1 + x^2} dx.
  • The differential of 1+x21 + x^2 is d(1+x2)=(1+x2)′dx=2xdxd(1 + x^2) = (1 + x^2)' dx = 2x dx.
  • Rewrite the integral as ∫1+x2⋅2xdx=∫1+x2d(1+x2)\int \sqrt{1 + x^2} \cdot 2x dx = \int \sqrt{1 + x^2} d(1 + x^2).
  • Let u=1+x2u = 1 + x^2 be the substitution, then ∫1+x2d(1+x2)=∫udu\int \sqrt{1 + x^2} d(1 + x^2) = \int \sqrt{u} du.
  • Integrating ∫udu\int \sqrt{u} du gives 23u3/2+C=23(1+x2)3/2+C\frac{2}{3} u^{3/2} + C = \frac{2}{3} (1 + x^2)^{3/2} + C.