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}{...

Tangent & Normal - Notes, Concept and All Important Formula

TANGENT & NORMAL 1. TANGENT TO THE CURVE AT A POINT: The tangent to the curve at 'P' is the line through P whose slope is limit of the secant's slope as \(Q \rightarrow P\) from either side. All Chapter Notes, Concept and Important Formula 2. NORMAL TO THE CURVE AT A POINT: A line which is perpendicular to the tangent at the point of contact is called normal to the curve at that point. 3. THINGS TO REMEMBER : (a) The value of the derivative at \(\mathrm{P}\left(\mathrm{x}_{1}, \mathrm{y}_{1}\right)\) gives the slope of the tangent to the curve at P. Symbolically \(\left.\mathrm{f}^{\prime}\left(\mathrm{x}_{1}\right)=\dfrac{\mathrm{dy}}{\mathrm{dx}}\right]_{\left(\mathrm{x}_{1}, \mathrm{y}_{1}\right)}=\) Slope of tangent at \(\mathrm{P}\left(\mathrm{x}_{1}, \mathrm{y}_{1}\right)=\mathrm{m}\) (say). (b) Equation of tangent at \(\left(x_{1}, y_{1}\right)\) is \(\left.y-y_{1}=\dfrac{d y}{d x}\right]_{\left(x_{1}, y_{1}\right)}\left(x-x_{1}\right)\) (c) Equation of...

Sequence And Series - Notes, Concept and All Important Formula

SEQUENCE & SERIES 1. ARITHMETIC PROGRESSION (AP) : AP is sequence whose terms increase or decrease by a fixed number. This fixed number is called the common difference . If ‘a’ is the first term & ‘d’ is the common difference, then AP can be written as a, a + d, a + 2d, ..., a + (n – 1) d, ... (a) \(n^{\text {th }}\) term of this AP \(\boxed{T_{n}=a+(n-1) d}\) , where \(d=T_{n}-T_{n-1}\) (b) The sum of the first \(n\) terms : \(\boxed{S_{n}=\frac{n}{2}[2 a+(n-1) d]=\frac{n}{2}[a+\ell]}\) , where \(\ell\) is the last term. (c) Also \(n ^{\text {th }}\) term \(\boxed{T _{ n }= S _{ n }- S _{ n -1}}\) Note: (i) Sum of first n terms of an A.P. is of the form \(A n^{2}+B n\) i.e. a quadratic expression in n, in such case the common difference is twice the coefficient of \(n ^{2}\) . i.e. 2A (ii) \(n ^{\text {th }}\) term of an A.P. is of the form \(An + B\) i.e. a linear expression in \(n\) , in such case the coefficient of \(n\) is the common difference of the ...

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