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Algebra of Integrals

Algebra of Integrals

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 »  Indefinite integrals under Basic Arithmetic Operations
    →  `int audx = a int udx`
    →  `int (u+v)dx = int udx + int vdx`
    →  `int (u-v)dx = int udx - int vdx`

Properties of Indefinite Integrals

plain and simple summary

nub

plain and simple summary

nub

dummy

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.

simple steps to build the foundation

trek

simple steps to build the foundation

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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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Starting on learning "". 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.

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

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.

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`
  • 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).

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

comprehensive information for quick review

Jogger

comprehensive information for quick review

Jogger

dummy

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`



           

practice questions to master the knowledge

Exercise

practice questions to master the knowledge

Exercise

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

  • `(2x^3)/3`
  • `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`

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`

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

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Indefinite integral or anti-derivative is defined as integral f of x dx, = g of x +c, where g of x is found by aggregate of change represented by limit of summation. ;; Or g of x is found such that derivative of g of x = f of x. ;; or g of x is the area under the curve f of x, between 0 and x. To understand the properties of the 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 of x = a multiplied u of x. ;; integral v d x ;; is derived to be a times integral u d x. ;; What does the above prove.
integral;a;v;d;x
integral of a v d x = a integral v d x
2
both;above
The answer is "both the above"
Intuitive understanding for the given result is : aggregate of change multiplies when the function is multiplied. or Area under the curve multiples when the y values of curve is multiplied.
Given integral y d x = 2 x squared and v = y by 5, what is integral v d x.
cube;
2 x cube by 3
5;squared
2 by 5 x squared
The answer is "2 by 5 x squared"
Finding integral of sum or difference is given by first principles. What does the above prove.
plus;equals;+;plus
integral of; (u plus v) equals integral of u ;plus; integral of v
sum;difference
integral of a sum or difference is the sum or difference of integrals.
minus;equals;minus;-
integral of; (u minus v) equals integral of u ;minus; integral of v
all;above
all the above
The answer is "all the above".
Intuitive understanding for given result is: aggregate of change adds or subtracts when the function is added or subtracted. or area under the curves add when the functions are added.
Given integral u dx = sine x and integral v d x = x + 20, what is integral u + v dx.
1
2
The answer is "sine 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, et cetera, will be taken up in due course.
integral of a Multiple: integral a u d x = a, integral u d x. ;; Integral of Sum or Difference is the sum or difference of the integrals.

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