What is the integration of log (1+x^2)?

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{...

Straight Line - Notes, Concept and All Important Formula

STRAIGHT LINE Table Of Contents 1. RELATION BETWEEN CARTESIAN CO-ORDINATE & POLAR CO-ORDINATE SYSTEM If

$(x, y)$ are Cartesian co-ordinates of a point

$P$ , then :

$x=r \cos \theta$ ,

$y=r \sin \theta$ and

$r=\sqrt{x^{2}+y^{2}}, \quad \theta=\tan ^{-1}\left(\dfrac{y}{x}\right)$ All Chapter Notes, Concept and Important Formula 2. DISTANCE FORMULA AND ITS APPLICATIONS : If

$\mathrm{A}\left(\mathrm{x}_{1}, \mathrm{y}_{1}\right)$ and

$\mathrm{B}\left(\mathrm{x}_{2}, \mathrm{y}_{2}\right)$ are two points, then

$\mathbf{A B=\sqrt{\left(x_{2}-x_{1}\right)^{2}+\left(y_{2}-y_{1}\right)^{2}}}$ Note : (i) Three given points

$A, B$ and

$C$ are collinear, when sum of any two distances out of

$\mathrm{AB}, \mathrm{BC}, \mathrm{CA}$ is equal to the remaining third otherwise the points will be the vertices of triangle. (ii) Let

$A, B, C \& D$ be the four given points in a plane. Then the quadrilateral will be: (a) Square if \(A B=B C=C D=D...

MATHEMATICAL WORLD

Search This Blog