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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

Properties of Indefinite Integrals

In this page, the integrals of functions that are given as arithmetic operation of multiple functions is discussed. The arithmetic operations are multiplication by a constant, addition, and subtraction.



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Indefinite integral or anti-derivative is defined as `int f(x)dx = g(x) +c`, where

 • `g(x)` is found by aggregate-of-change represented by `lim_(n->oo) sum_(i=1)^n f((i x)/n) xx x/n`

 • `g(x)` is found such that `d/(dx) g(x) = f(x)`.

 • `g(x)` is the area under the curve `f(x)` between `0` and `x`. To understand the properties of indefinite integrals one of these is used.

 • aggregate of change provides rigorous mathematical proof.

 • anti-derivatives provide proof derived from properties of derivatives.

 • area under the curve provide the geometrical methods.

A proof given in one of these three can be verified in other.

Integral of a scalar multiple of a function: Given `v(x)=color(deepskyblue)au(x)`.

`int v dx `

`quad quad = lim_(n->oo) sum_(i=1)^n (v((i x)/n) x/n`

`quad quad = lim_(n->oo) sum_(i=1)^n (color(deepskyblue)au((i x)/n) x/n`

with continuity and integrability conditions on `u`
`quad quad = color(deepskyblue)a lim_(n->oo) sum_(i=1)^n (u((i x)/n) x/n`

`quad quad = color(deepskyblue)a int u dx`

what does the above prove?

  • `int a v dx = a int v dx `
  • integral of a multiple of a function is multiple of the integral of the function
  • Both the above
  • Both the above

The answer is "both the above"

Intuitive understanding for
`int au dx; = a int u dx`

 • aggregate of change multiplies when the function is multiplied by a constant.area under the curve for a u(x)  • area under the curve multiplies when the `y` values of curve is multiplied.

Solved Exercise Problem:

Given `int y dx = 2x^2` and `v=y/5`, what is `int v dx`?

  • `(2x^3)/3`
  • `2/5 x^2`
  • `2/5 x^2`

The answer is "`2/5 x^2`"

`int v dx`
`= int y/5 dx`
`= 1/5 int y dx`
`=1/5 xx 2x^2`
`=2/5 x^2`

Finding integral of sum or difference.

`int (u+v)dx`

`quad = lim_(n->oo) sum_(i=1)^n ``(u((i x)/n)+-v((i x)/n))``xx x/n`


`quad = lim_(n->oo) sum_(i=1)^n ``u((i x)/n)xx x/n +- v((i x)/n)``xx x/n`


with continuity and integrability conditions on `u` and `v`
`quad = lim_(n->oo) sum_(i=1)^n ``u((i x)/n)xx x/n`

`+- lim_(n->oo) sum_(i=1)^n ``v((i x)/n)xx x/n`


`quad quad = int u dx +- int v dx`

What does the above prove? `

  • `int (u+v) dx = int u dx + int v dx`
  • `int (u+v) dx = int u dx + int v dx`
  • integral of a sum or difference is the sum or difference of integrals.
  • `int (u-v) dx = int u dx - int v dx`
  • all the above

The answer is "all the above".

Intuitive understanding for
`int (u+-v) dx = int u dx +- int v dx`

 • aggregate of change adds (subtracts) when the function is added(subtracted).integral of addition or subtraction of functions  • Areas under the curves add (or subtract) when the functions are added (or subtracted).

Solved Exercise Problem:

Given `int u dx = sinx` and `int v dx =x+20`, what is `int (u+v) dx`?

  • `(sin x)(x+20)`
  • `sin x + x + 20`
  • `sin x + x + 20`

The answer is "`sin x + x + 20`"

Integral of a multiple of a function is multiple of the integral of the function.

Integral of a sum or difference is sum or difference of integrals.

Note: Other algebraic operations (like multiplication, function of function, etc.) will be taken up in due course.

Integral of a Multiple:
`int (au)dx = a int u dx`

Integral of Sum or Difference:
`int (u+v) dx = int u dx + int v dx`

`int (u-v) dx = int u dx - int v dx`

                            
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