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

Read in the blogs more about the unique learning experience at nubtrek.continue

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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`

*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*

trek

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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.

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 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). • 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`"

*your progress details*

Progress

*About you*

Progress

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.