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Various Forms and Results of Integrals

Various Forms and Results of Integrals

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 »  Understanding Complexity of Integration Methods
functions given as multiplication or division of sub-functions.

 → Integration by Identities `Pi_(i=1)^n g_i(x) = sum_(i=1)^m h_i(x)`

 → Integration by Substitution where `f(x) = g(h(x))h′(x)dx`

 → Integration by Parts where `int g(x)h(x)dx = g(x)int h(x)dx - int [int h(x) dx] g′(x)dx`

 → Special cases of Integration by Parts where `(g′(x))′ = -g(x)`

 → Special cases of Integration by Parts where `g(x) = e^x` and `h(x) = r(x)+r′(x)`

 → Integration of Division by Polynomial: `(r(x))/(q(x)) = t(x) + (p(x))/(q(x))`, where order of `p(x)` is smaller than order of `q(x)`.

 → Integration by Partial fractions `p(x)/(Pi_(i=1)^n g_i(x)) = sum_(i=1)^m 1/(h_i(x))` where `1/(h_i(x))` is integrable.

Methods of Integration

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In this page, the standard forms of integration with multiplication and division are explained. Multiplication or division in integrand can be converted into something for which integration can be worked out.


Keep tapping on the content to continue learning.
Starting on learning "". In this page, the standard forms of integration with multiplication and division are explained. Multiplication or division in integrand can be converted into something for which integration can be worked out.

We have learned the standard results of anti-derivatives as inverse of derivative. In that only few of the expressions are not covered. For example, consider `int tan x dx`. Has this been detailed yet?

  • Yes, it has already been covered
  • No, not yet.

The answer is "No, not yet". We will solve this problem in due course of this lesson.

The point being, there are several functions for which finding result of integration is not explained yet in this course.

It is also noted here that not all functions have an anti-derivative in the form of a function. For example, `int e^(-x^2)dx` is widely used in other scientific areas of studies, but the integral has no function-form. How do they find integral of this?

  • by numerical integration or a table
  • there is an anti-derivative possible

The answer is "by numerical integration or a table". This part is not covered in this course.

Though there are several functions for which integration cannot be worked out as a function, there are several other forms of functions, that can be integrated with some algebraic manipulations. The objective of this page is to provide some insights to that.

Usually, functions are made of sub-functions of standard forms with operations.

 • standard forms of functions: `x^n`, trigonometric functions, inverse trigonometric functions, exponents, and logarithm

 • Standard operations:

     addition of functions `f(x)+g(x)`,

     subtraction of functions `f(x)-g(x)`,

     multiplication of functions `f(x) xx g(x)`,

     division `f(x)/g(x)`,

     multiplication or division by a constant `af(x)`,

     exponents `f(x)^a`,

     roots `f(x)^(1/n)`,

     scaling `f(ax)`,

     shifting `f(x+a)`, and

     function of function `f(g(x))` Which of these standard operations are covered in algebra of integrals?

  • addition, subtraction, and multiplication by a constant
  • none of the operations

The answer is "addition, subtraction, and multiplication by a constant" were covered in algebra of integrals.

`int [af(x)+-g(x)]dx ``=a int f(x)dx +- int g(x)dx`

In this lesson, we will focus on integration methods of functions with multiplication or division of sub-functions. Assume `g_i(x)` is a standard function for which anti-derivative is known. If a function is given as `f(x) = Pi_(i=1)^n g_i(x)`, that is the function equals multiplication of several sub-functions. How to work out the integral of the function?

To solve this, several patterns and their properties are understood for different cases. Based on the properties, methods to handle the functions to work out the integration are provided. These are

 • Integration by Identities `Pi_(i=1)^n g_i(x) = sum_(i=1)^m h_i(x)`

 • Integration by Substitution where `f(x) = g(h(x))h′(x)dx`

 • Integration by Parts where `int g(x)h(x)dx = g(x)int h(x)dx - int [int h(x) dx] g′(x)dx`

 • Special cases of Integration by Parts where `(g′(x))′ = -g(x)`

 • Special cases of Integration by Parts where `g(x) = e^x` and `h(x) = r(x)+r′(x)`

 • Integration of Division by Polynomial: `(r(x))/(q(x)) = t(x) + (p(x))/(q(x))`, where order of `p(x)` is smaller than order of `q(x)`.

 • Integration by Partial fractions `p(x)/(Pi_(i=1)^n g_i(x)) = sum_(i=1)^m 1/(h_i(x))` where `1/(h_i(x))` is integrable.

Let us restrict to this set of integrations.

This is an overview of the whole lesson.

The underlying pattern in this whole lesson is -- when we cannot work out integration of an integrand, convert the multiplication in integrand into something for which integration can be worked out. Unless you see this underlying pattern and connect the methods of integration, the results will be quite tedious and voluminous to learn.

comprehensive information for quick review

Jogger

comprehensive information for quick review

Jogger

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practice questions to master the knowledge

Exercise

practice questions to master the knowledge

Exercise

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Progress

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Progress

We have learned the standard results of anti-derivatives as inverse of derivative. In that only few of the expressions are not covered. For example, consider integral tan x dx. Has this been detailed yet.
s;yes;already;covered
Yes, it has already been covered
no;not;yet
No, not yet.
The answer is "No, not yet". We will solve this problem in due course of this lesson.
The point being, there are several functions for which finding result of integration is not explained yet in this course. ;; It is also noted here that not all functions have an anti-derivative in the form of a function. For example, integral e power, minus x squared, dx, is widely used in other scientific areas of studies, but the integral has no function-form. How do they find integral of this
numerical;integration;table
by numerical integration or a table
anti;derivative;possible
there is an anti-derivative possible
The answer is "by numerical integration or a table". This part is not covered in this course.
Though there are several functions for which integration cannot be worked out as a function, there are several other forms of functions, that can be integrated with some algebraic manipulations. The objective of this page is to provide some insights to that. Usually, functions are made of sub-functions of standard forms with operations. Standard forms of functions and standard operations are listed. ;; Which of these standard operations are covered in algebra of integrals?
addition;subtraction;multiplication;constant
addition, subtraction, and multiplication by a constant
none;operations
none of the operations
The answer is "addition, subtraction, and multiplication by a constant" were covered in algebra of integrals. It is summarized.
In this lesson, we will focus on integration methods of functions with multiplication or division of sub-functions. Assume g i of x is a standard function for which anti-derivative is known. If a function is given as f of x multiplication from i = 1 to n, g i of x. That is the function equals multiplication of several sub-functions. How to work out the integral of the function? To solve this, several patterns and their properties are understood for different cases. Based on the properties, methods to handle the functions to work out the integration are provided. This is an overview of the whole lesson.
The underlying pattern in this whole lesson is -- when we cannot work out integration of an integrand, convert the multiplication in integrand into something for which integration can be worked out. Unless you see this underlying pattern and connect the methods of integration, the results will be quite tedious and voluminous to learn.

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