Mathematics - All Important Notes, Formula and Concept

Indefinite Integration - Notes, Concept and All Important Formula

INDEFINITE INTEGRATION If f & F are function of

$x$ such that

$F^{\prime}(x)$

$=f(x)$ then the function

$F$ is called a PRIMITIVE OR ANTIDERIVATIVE OR INTEGRAL of

$\mathrm{f}(\mathrm{x})$ w.r.t.

$\mathrm{x}$ and is written symbolically as

$\displaystyle \int \mathrm{f}(\mathrm{x}) \mathrm{d} \mathrm{x}$

$=\mathrm{F}(\mathrm{x})+\mathrm{c} \Leftrightarrow \dfrac{\mathrm{d}}{\mathrm{dx}}\{\mathrm{F}(\mathrm{x})+\mathrm{c}\}$

$=\mathrm{f}(\mathrm{x})$ , where

$\mathrm{c}$ is called the constant of integration. Note : If

$\displaystyle \int f(x) d x$

$=F(x)+c$ , then

$\displaystyle \int f(a x+b) d x$

$=\dfrac{F(a x+b)}{a}+c, a \neq 0$ All Chapter Notes, Concept and Important Formula 1. STANDARD RESULTS : (i)

$\displaystyle \int(a x+b)^{n} d x$

$=\dfrac{(a x+b)^{n+1}}{a(n+1)}+c ; n \neq-1$ (ii)

$\displaystyle \int \dfrac{d x}{a x+b}$

$=\dfrac{1}{a} \ln|a x+b|+c$ (iii)

$\displaystyle \int e^{\mathrm{ax}+b} \mathrm{dx}$ \(=\dfrac{1}{...

Logarithm - Notes, Concept and All Important Formula

LOGARITHM LOGARITHM OF A NUMBER : The logarithm of the number

$\mathrm{N}$ to the base '

$\mathrm{a}$ ' is the exponent indicating the power to which the base 'a' must be raised to obtain the number

$\mathrm{N}$ . This number is designated as

$\log _{\mathrm{a}} \mathrm{N}$ . (a)

$\log _{a} \mathrm{~N}=\mathrm{x}$ , read as

$\log$ of

$\mathrm{N}$ to the base

$\mathrm{a} \Leftrightarrow \mathrm{a}^{\mathrm{x}}=\mathrm{N}$ . If

$a=10$ then we write

$\log N$ or

$\log _{10} \mathrm{~N}$ and if

$\mathrm{a}=e$ we write

$\ln N$ or

$\log _{e} \mathrm{~N}$ (Natural log) (b) Necessary conditions :

$\mathrm{N}> \,\,0 ; \,\, \mathrm{a}> \,\,0 ; \,\, \mathrm{a} \neq 1$ (c)

$\log _{a} 1=0$ (d)

$\log _{a} a=1$ (e)

$\log _{1 / a} a=-1$ (f)

$\log _{a}(x . y)=\log _{a} x+\log _{a} y ; \,\, x, y> \,\,0$ (g) \(\log _{a}\left(\dfrac{\mathrm{x}}{y}\right)=\log _{\mathrm{a}} \mathrm{x}-\log _{\mathrm{a}} \mathrm{y} ; \,\, \mathrm{...

Vectors - Notes, Concept and All Important Formula

VECTORS 1. Vectors Physical quantities are broadly divided in two categories viz (a) Vector Quantities & (b) Scalar quantities. (a) Vector quantities: Any quantity, such as velocity, momentum, or force, that has both magnitude and direction and for which vector addition is defined and meaningful; is treated as vector quantities. (b) Scalar quantities: A quantity, such as mass, length, time, density or energy, that has size or magnitude but does not involve the concept of direction is called scalar quantity. All Chapter Notes, Concept and Important Formula 2. REPRESENTATION : Vectors are represented by directed straight line segment magnitude of

$\overrightarrow{ a }=|\overrightarrow{ a }|=$ length

$PQ$ direction of

$\overrightarrow{a}=P$ to

$Q$ 3. Types of Vectors (a) ZERO VECTOR OR NULL VECTOR : A vector of zero magnitude i.e. which has the same initial & terminal point is called a ZERO VECTOR. It is denoted by

$\overrightarrow{O}$ . (b) UNIT VECTOR : A v...

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