Substitution Method
For a differentiable function , if you let , the integral transforms as follows:
Indefinite Integral of the Form
If you let , then , and by substitution:
Indefinite Integral of Rational Functions
When integrating rational functions, expressing the function as a sum or difference of simpler rational functions can be convenient.
When the Degree of the Numerator Degree of the Denominator:
Divide the numerator by the denominator, then integrate the resulting expression.
Example:
When the Denominator Can Be Factored, and the Degree of the Numerator Degree of the Denominator:
Express the integrand as partial fractions before integrating.
Example:
Definite Integration Using Substitution
For a function that is continuous on the closed interval , and for a differentiable function where and , if is continuous over the interval , the definite integral transforms as:
Substitution Using Trigonometric Functions
For integrals involving (with ):
Use the substitution where .
For integrals involving (with )
Use the substitution where .