Definition of Factorials

The factorial function multiplies a series of descending positive integers from n down to 1

Definition of the Gamma Function

The gamma function extends the concept of factorials for non-integer values and is denoted as Γ(n)

Relationship between Factorials and the Gamma Function

The gamma function is related to factorials through the relationship Γ(n+1) = n! for all positive integers n

Definition of the Power Rule

The power rule states that the integral of ax^n with respect to x is (a/(n+1))x^(n+1) plus a constant of integration C, for any real number n different from -1

Antiderivative Concept

The power rule is a direct consequence of the antiderivative concept, which reverses differentiation

Examples of Using the Power Rule

The power rule can be used to integrate polynomial functions, such as 3x^3, which yields (3/4)x^4 + C

Substitution

Substitution, also known as u-substitution, is a method used for integrating composite functions

Integration by Parts

Integration by parts is a technique used for integrating products of functions

Partial Fraction Decomposition

Partial fraction decomposition is used for integrating rational functions that can be expressed as a sum of simpler fractions

Integrating Inverse Functions

Integrals of inverse functions can be expressed as xf^-1(x) - F(x) + C, where F is an antiderivative of f

Trigonometric Integrals

Trigonometric integrals may require the use of identities or substitutions to simplify the integral before applying standard techniques

Differentiating Under the Integral Sign

Differentiating under the integral sign, also known as the Leibniz integral rule, is a method used for evaluating difficult integrals