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Algebra of Integrals

Algebra of Integrals

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 »  Understanding Algebra of Integrals
how integral applies to a function given as algebraic operations of several functions

    →  addition and subtraction

    →  product and division

    →  function-of-function

    →  parametric form of function

Understanding Algebra of Integration

plain and simple summary

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plain and simple summary

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In this page, what is algebra of integrals and conditions under which it is applicable are discussed.


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Starting on learning "Understanding Algebra of Integration". In this page, what is algebra of integrals and conditions under which it is applicable are discussed.

What does the title "Algebra of Integration" or "Algebra of Integrals" mean?

  • Properties to find integrals of functions given as algebraic operations of several functions
  • application of integration

The answer is "Properties to find integrals of functions given as algebraic operations of several functions"

The mathematical operations are

 •  addition and subtraction `u(x) +- v(x)`

 •  multiple of a function `a u(x)`

 •  multiplication and division `u(x)v(x)` and `(u(x))/(v(x))`

 •  powers and roots `[u(x)]^n` and `[u(x)]^(1/n)`

 •  composite form of functions `v (u(x))`

 •  parametric form of functions `v=f(r) ; u=g(r)`

Given that `f(x) = u(x)***v(x)` where `***` is one of the arithmetic or function operations.

Will there be any relationship between the integrals of the functions `int u(x) dx` ; `int v(x) dx` and the integral of the result `int f(x) dx`?

Algebra of integration analyses this and provides the required knowledge.

Note: In deriving the results, the functions are assumed to be continuous and integrable at the range of interest. For specific functions at specific intervals, one must check for the continuity and the integrability before using the algebra of integrals.

For example, consider
`u(x) = x^2`
`v(x) = sin x`
`f(x) = x^2 sin x`

From the standard results, it is known that
`int x^2 dx = x^3/3 + c` and
`int sin x dx = -cos x + c`.
What is `int x^2 sin x dx`?

In this particular example multiplication is considered. Instead of multiplication, one of the arithmetic or function operations may be considered too.

The algebra of integrals analyses this and provides the required knowledge to solve.

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comprehensive information for quick review

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Exercise

practice questions to master the knowledge

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What does the title : Algebra of Integration, or, Algebra of Integrals mean.
find;given;algebraic;several
Properties to find integrals of functions given as algebraic operations of several functions
application
application of integration
The answer is "Properties to find integrals of functions given as algebraic operations of several functions"
The operations are, addition and subtraction; multiple of a function; multiplication and division ; powers and roots; composite form of function ; parametric form of functions.
Given that f of x equals u of x star v of x. Where star is one of the arithmetic, or, function operations. Will there be any relationship between the integral of the functions, integral u of x d x ; integral v of x d x, and the integral of the result integral f of x d x? Algebra of integration analyses this and provides the required knowledge.
For example, consider u of x equals x squared, v of x equals sine x , f of x equals x squared sine x. From the standard results of integrals, it is known that integral x squared = x cubed by 3 + c and integral sine x dx = negative cos x . What is integral x squared sine x dx? In this particular example multiplication is considered. Instead of multiplication, one of the arithmetic or function operations may be considered too. The algebra of integrals analyses this and provides the required knowledge to solve.

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