Indefinite Integrals of the Function (where is a real number)

For any real number
  1. If :
  2. If :

Indefinite Integrals of Exponential Functions

  2. for and
For expressions of the form (where and is a real number), use the exponential law to rewrite it as , then integrate:


Indefinite Integrals of Trigonometric Functions

For and , use the identities:
to rewrite the expressions before integrating.

Definite Integrals

Definite Integral

  1. If is continuous on the closed interval , and is an antiderivative of , then the definite integral from to is given by:



  3. If is continuous on the interval , then:
    for

Properties of Definite Integrals

If and are continuous on an interval containing the real numbers , , and , then:
  1. where is a constant

Definite Integrals of Even and Odd Functions

If is continuous on the closed interval :
  1. If , i.e., is an even function, then:
  2. If , i.e., is an odd function, then:

Definite Integrals of Periodic Functions

For a continuous function with period :

Functions Defined by Definite Integrals

  1. Differentiation of Functions Defined by Definite Integrals
    1. where is a constant
    2. where is a constant

  2. Limits of Functions Defined by Definite Integrals