You get a bonus - 1 coin for daily activity. Now you have 1 coin

3.1.2. Integral table. Definition and formulas

Lecture



  3.1.2.  Integral table.  Definition and formulas

Integration is one of two basic operations in mathematical analysis. Unlike the differentiation operation, the integral of an elementary function may not be an elementary function. For example, it follows from the Liouville theorem that the integral of   3.1.2.  Integral table.  Definition and formulas not an elementary function. Tables of known primitives are often very useful, although now they lose their relevance with the advent of computer algebra systems. This page lists the most common primitives.

  3.1.2.  Integral table.  Definition and formulas used as an arbitrary integration constant that can be determined if the value of the integral is known at some point. Each function has an infinite number of antiderivatives.

Content

  • 1Rules for integrating functions
  • 2 Integrals of elementary functions
    • 2.1Rational functions
    • 2.2Logarithms
    • 2.3Exponential Functions
    • 2.4Rational functions
    • 2.5Trigonometric functions
    • 2.6 Hyperbolic functions

Function Integration Rules

  3.1.2.  Integral table.  Definition and formulas

  3.1.2.  Integral table.  Definition and formulas

  3.1.2.  Integral table.  Definition and formulas

  3.1.2.  Integral table.  Definition and formulas

  3.1.2.  Integral table.  Definition and formulas

Integrals of elementary functions

Rational functions

List of integrals of rational functions

  3.1.2.  Integral table.  Definition and formulas (antiderivative from zero is a constant, in any limits of integration the integral from zero is zero)

}   3.1.2.  Integral table.  Definition and formulas

  3.1.2.  Integral table.  Definition and formulas

  3.1.2.  Integral table.  Definition and formulas

Evidence

Make a replacement   3.1.2.  Integral table.  Definition and formulas get

  3.1.2.  Integral table.  Definition and formulas

  3.1.2.  Integral table.  Definition and formulas ("High logarithm")

Logarithms

List of integrals of logarithmic functions

  3.1.2.  Integral table.  Definition and formulas

  3.1.2.  Integral table.  Definition and formulas

  3.1.2.  Integral table.  Definition and formulas

Exponential Functions

List of integrals of exponential functions

  3.1.2.  Integral table.  Definition and formulas

  3.1.2.  Integral table.  Definition and formulas

Irrational functions

List of integrals of irrational functions

Differential bean

  3.1.2.  Integral table.  Definition and formulas

  3.1.2.  Integral table.  Definition and formulas

  3.1.2.  Integral table.  Definition and formulas

  3.1.2.  Integral table.  Definition and formulas ("Long logarithm")

  3.1.2.  Integral table.  Definition and formulas

Evidence

Let be   3.1.2.  Integral table.  Definition and formulas , suppose also that   3.1.2.  Integral table.  Definition and formulas . Let's use hyperbolic functions, we will make replacement   3.1.2.  Integral table.  Definition and formulas

  3.1.2.  Integral table.  Definition and formulas

But

  3.1.2.  Integral table.  Definition and formulas   3.1.2.  Integral table.  Definition and formulas   3.1.2.  Integral table.  Definition and formulas

therefore

  3.1.2.  Integral table.  Definition and formulas

From here, including the logarithm of the denominator of the last fraction in the constant C, we get

  3.1.2.  Integral table.  Definition and formulas

If a   3.1.2.  Integral table.  Definition and formulas then replacing   3.1.2.  Integral table.  Definition and formulas we reduce the integral to the case already considered. If   3.1.2.  Integral table.  Definition and formulas , then we make a replacement   3.1.2.  Integral table.  Definition and formulas and carry out arguments similar to the case considered [1].

Trigonometric functions

list of integrals of trigonometric functions , list of integrals of inverse trigonometric functions

  3.1.2.  Integral table.  Definition and formulas

  3.1.2.  Integral table.  Definition and formulas

  3.1.2.  Integral table.  Definition and formulas

Evidence

  3.1.2.  Integral table.  Definition and formulas

  3.1.2.  Integral table.  Definition and formulas

Evidence

  3.1.2.  Integral table.  Definition and formulas

  3.1.2.  Integral table.  Definition and formulas

  3.1.2.  Integral table.  Definition and formulas

  3.1.2.  Integral table.  Definition and formulas

  3.1.2.  Integral table.  Definition and formulas

  3.1.2.  Integral table.  Definition and formulas

  3.1.2.  Integral table.  Definition and formulas

  3.1.2.  Integral table.  Definition and formulas

  3.1.2.  Integral table.  Definition and formulas

  3.1.2.  Integral table.  Definition and formulas

  3.1.2.  Integral table.  Definition and formulas

  3.1.2.  Integral table.  Definition and formulas

Hyperbolic functions

List of integrals of hyperbolic functions

  3.1.2.  Integral table.  Definition and formulas

  3.1.2.  Integral table.  Definition and formulas

  3.1.2.  Integral table.  Definition and formulas

  3.1.2.  Integral table.  Definition and formulas

  3.1.2.  Integral table.  Definition and formulas

  3.1.2.  Integral table.  Definition and formulas

  3.1.2.  Integral table.  Definition and formulas

also   3.1.2.  Integral table.  Definition and formulas

also   3.1.2.  Integral table.  Definition and formulas

  3.1.2.  Integral table.  Definition and formulas

Proof of

Proof of formula   3.1.2.  Integral table.  Definition and formulas :

  3.1.2.  Integral table.  Definition and formulas

Proof of formula   3.1.2.  Integral table.  Definition and formulas :   3.1.2.  Integral table.  Definition and formulas .

Proof of formula   3.1.2.  Integral table.  Definition and formulas :

  3.1.2.  Integral table.  Definition and formulas


Comments


To leave a comment
If you have any suggestion, idea, thanks or comment, feel free to write. We really value feedback and are glad to hear your opinion.
To reply

Mathematical analysis. Integral calculus

Terms: Mathematical analysis. Integral calculus