Formula of Integral Calculus | Basic Integration Formulas | ইন্টিগ্রেশন ক্যালকুলাস এর সূত্রসমূহ

✪➤ Formula of Integral Calculus | Basic Integration Formulas
☞ The remark that integration is (almost) an inverse to the operation of differentiation means that if
$\frac {d}{dx}f(x) = g(x)$       [Differentiation]

then,  $\int g(x) dx = f(x) + c$        [Integration]

✪Most Used Formula:

$\int dx= x+ c$

$\int x dx = \frac{x^2}{2}+c$

$\int x^n dx = \frac{x^n+1}{n+1}+c$

$\int e^x dx = e^x+c$

$\int e^{mx} dx$ = $ \Large \frac {e^{mx}}{m}+c$

$\int e^{-mx} dx$ = $\Large \frac {e^{-mx}}{-m}+c$

$\int \frac{1}{x} dx= log x+c$

$\int \sin x\; dx= -\cos x\;+c$

$\int \sin mx\; dx$= $\Large \frac {-cos mx}{m}+c$

$\int \cos x\; dx= \sin x\;+c$

$\int \cos mx\; dx$= $\Large \frac {sin mx}{m}+c$

$\int sec^2 x dx= \tan x\;+c$
$\int cosec^2 x dx= -\cot x\;+c$
$\int \sec x\; \tan x\; dx= \sec x\;+c$
$\int cosecx \cot x\; dx= -cosec x+c$

$\large \int \frac {1}{1+x^2} dx$= $tan^{-1} x+c$

$\large \int \frac {1}{\sqrt{1-x^2}} dx$ =$ sin^{-1} x+c$

$\int \frac {f'(x)}{f(x)} dx= log f(x)+c$

$\int uv$ $dx$= $u \int v dx - \int[ \frac {d}{dx}(u)$ $\int v$dx$ ]$dx + c


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Number Systems

Number: A number is a mathematical object used to count, measure, and label. The original examples are the natural numbers 1, 2, 3, 4 and so forth.

Main classification:

The major categories of numbers are as follows:

Main number systems

${\displaystyle \mathbb {N} }$	Natural	0, 1, 2, 3, 4, 5, ... or 1, 2, 3, 4, 5, ... ${\displaystyle \mathbb {N} _{0}}$ or ${\displaystyle \mathbb {N} _{1}}$ are sometimes used.
${\displaystyle \mathbb {Z} }$	Integer	..., −5, −4, −3, −2, −1, 0, 1, 2, 3, 4, 5, ...
${\displaystyle \mathbb {Q} }$	Rational	a/b where a and b are integers and b is not 0
${\displaystyle \mathbb {R} }$	Real	The limit of a convergent sequence of rational numbers
${\displaystyle \mathbb {C} }$	Complex	a + bi where a and b are real numbers and i is a formal square root of −1

Natural numbers:

The most familiar numbers are the natural numbers (sometimes called whole numbers or counting numbers): 1, 2, 3, and so on. 

At present, In the base 10 numeral system, in almost universal use today for mathematical operations, the symbols for natural numbers are written using ten digits: 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9. The radix or base is the number of unique numerical digits, including zero, that a numeral system uses to represent numbers (for the decimal system, the radix is 10).

Integers:

An integer (from the Latin integer meaning "whole") is a number that can be written without a fractional component. For example, 21, 4, 0, and −2048 are integers, while 9.75, ​5 1⁄2, and √2 are not.

Rational number:

A rational number is a number that can be expressed as a fraction with an integer numerator and a positive integer denominator. 

Negative denominators are allowed, but are commonly avoided, as every rational number is equal to a fraction with positive denominator.

The fraction m/n represents m parts of a whole divided into n equal parts. Two different fractions may correspond to the same rational number; for example 1/2 and 2/4 are equal, that is:

${\displaystyle {1 \over 2}={2 \over 4}.}$

In general,

${\displaystyle {a \over b}={c \over d}}$ if and only if ${\displaystyle {a\times d}={c\times b}.}$

Real numbers:
A real number is a value that represents a quantity along a line.The real numbers include all the measuring numbers. The symbol for the real numbers is R.

The real numbers include all the rational numbers, such as the integer −5 and the fraction 4/3, and all the irrational numbers, such as √2 (1.41421356..., the square root of 2, an irrational algebraic number).

Complex numbers:

A complex number is a number that can be expressed in the form a + bi, where a and b are real numbers, and i is a solution of the equation x² = −1, which is called an imaginary number because there is no real number that satisfies this equation. For the complex number a + bi, a is called the real part, and b is called the imaginary part.

Subclasses of the integers:

Even and odd numbers:

An even number is an integer that is "evenly divisible" by two, that is divisible by two without remainder; an odd number is an integer that is not even

Prime numbers:
A prime number is an integer greater than 1 that is not the product of two smaller positive integers. The first few prime numbers are 2, 3, 5, 7, and 11

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Formulas of Differential Calculus | Derivative Rules | ক্যালকুলাস এর সূত্রসমূহ

Formulas of Differential Calculus | Derivative Rules | ক্যালকুলাস এর সূত্রসমূহ

$\frac{d}{dx}(c)$  = 0                             [where, c is single constant]

$\frac{d}{dx}(cu)$ = c $\frac {d}{dx}$ (u)   [where, c is with a variable]

$\frac{d}{dx}$ (xⁿ) = nx^n-1

$\frac{d}{dx}$ $(\sqrt {x})= \frac {1}{2\sqrt{x}}$

$\frac{d}{dx}$ (e^x)= e^x

$\frac{d}{dx}$ (e^mx)= me^mx

$\frac{d}{dx}$ (e^-mx)= -me^-mx

$\frac{d}{dx}$ (log_a x)= $\frac {1}{x}$ log_ae

$\frac{d}{dx}$ (log x)= $\frac {1}{x}$

$\frac{d}{dx}$ (sin x)= cos x

$\frac{d}{dx}$ (sin mx)= m cos mx

$\frac{d}{dx}$ (cos x)= - sin x

$\frac{d}{dx}$ (cos mx)= - m sin mx

$\frac{d}{dx}$ (sec x)= sec x. tan x

$\frac{d}{dx}$ (cosec x)= -cosec x. cot x

$\frac{d}{dx}$ (tan x)= sec² x

$\frac{d}{dx}3$ (cot x)= -cosec² x

$\frac{d}{dx}$ (sin^-1) = $1  \over { \sqrt{1-x^2 }}$

$\frac{d}{dx}$ (cos^-1)= $-1  \over { \sqrt{1-x^2 }}$

$\frac{d}{dx}$ (tan^-1)= $\frac{1}{1+x^2}$

$\frac{d}{dx}$ (uv)= u $ \frac {d}{dx}$ (v) + v $\frac {d}{dx}$ (u)

$\frac{d}{dx}$ (uvw) = vw $\frac {d}{dx}$ (u) + uw $\frac {d}{dx}$ (v) + uv $\frac {d}{dx}$( w )

$\frac {d}{dx}$ ($\frac {u}{v})$ = $\frac{v\frac{\mathrm{d}}{\mathrm{d} x} ( u )-u\frac{\mathrm{d} }{\mathrm{d} x} ( v )}{v^{2}}$


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Algebra Formulas - বীজগণিত সুত্র সমূহ

✪✪✪ Algebraic Formulas | Algebraic Expression ✪✪✪

1.    (a + b)² = a² + 2ab + b²
2.    (a - b)² = a² - 2ab + b²
3.    (a + b)² = (a - b)² + 4ab
4.    (a - b)² = (a + b)² - 4ab
5.    (a + b+ c)² = a² + b² + c² + 2ab +2bc + 2ca
6.    (a + b+ c)² = a² + b² + c² - 2ab +2bc - 2ca
7.    (x + a)(x + b) = x² + (a + b)x + ab
8.    a² + b² = (a + b)² − 2ab
9.    a² + b² = (a - b)² + 2ab
10.    a² + b² = $\frac{(a + b)^{2} + (a - b)^{2}} {2}$
11.    2(a² + b² )= (a + b)² + (a - b)² 
12.  a² + b² + c² =(a + b+ c)² - 2(ab +bc + ca)
13. 2(ab +bc + ca) = (a + b+ c)² -  (a² + b² + c² )
14. a² ‑ b² = (a - b)² + 2ab
15.  a² ‑ b² = (a + b) (a - b)
16. 4ab = (a +b)² – (a – b)²
17. ab = $\frac {(a + b)^{2} - (a - b)^{2}} {4}$
18. (a + b)³ = a³ + 3a²b + 3ab² + b³
19. (a - b)³ = a³ - 3a²b + 3ab² - b³
20. (a³ + b³) = (a + b)³ – 3ab (a + b)
21. (a³ – b³) = (a - b)³ + 3ab (a - b)
22. (a³ + b³) = (a + b) (a² - ab + b²)
23. (a³ - b³) = (a - b) (a² + ab + b²)

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