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Thought-Process to Discover Knowledge

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Books and other education websites provide "matter-of-fact" knowledge. Instead, nubtrek provides a thought-process to discover knowledge.

In each of the topic, the outline of the thought-process, for that topic, is provided for learners and educators.

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mathsIntegral CalculusAlgebra of Integrals

Understanding Algebra of Integration

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



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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
  • 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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