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Thought-Process to Discover Knowledge

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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 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).  • 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`

slide-show version coming soon