Exploring the factorial and gamma functions reveals the mathematical operations extending factorials to non-integer values. The power rule for integration simplifies polynomial functions, while advanced techniques like substitution and integration by parts tackle complex integrals. Inverse function integration and trigonometric integrals require specific formulas and identities. Feynman's technique exemplifies the connection between differentiation and integration in calculus.
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The factorial function multiplies a series of descending positive integers from n down to 1
The gamma function extends the concept of factorials for non-integer values and is denoted as Γ(n)
The gamma function is related to factorials through the relationship Γ(n+1) = n! for all positive integers n
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
The power rule is a direct consequence of the antiderivative concept, which reverses differentiation
The power rule can be used to integrate polynomial functions, such as 3x^3, which yields (3/4)x^4 + C
Substitution, also known as u-substitution, is a method used for integrating composite functions
Integration by parts is a technique used for integrating products of functions
Partial fraction decomposition is used for integrating rational functions that can be expressed as a sum of simpler fractions
Integrals of inverse functions can be expressed as xf^-1(x) - F(x) + C, where F is an antiderivative of f
Trigonometric integrals may require the use of identities or substitutions to simplify the integral before applying standard techniques
Differentiating under the integral sign, also known as the Leibniz integral rule, is a method used for evaluating difficult integrals