Skip to main content

Definite Integration - Notes, Concept and All Important Formula

DEFINITE INTEGRATION

    1. (a) The Fundamental Theorem of Calculus, Part 1:

    If \(\mathrm{f}\) is continuous on \([\mathrm{a}, \mathrm{b}]\), then the function \(\mathrm{g}\) defined by

    \(g(x)=\displaystyle \int_{a}^{x} f(t) d t,\)  \(  a \leq x \leq b\) is continuous on \([\mathrm{a}, \mathrm{b}]\) and differentiable on \((\mathrm{a}, \mathrm{b})\), and \(g^{\prime}(\mathrm{x})=\mathrm{f}(\mathrm{x})\).

    (b) The Fundamental Theorem of Calculus, Part 2:

    If f is continuous on \([a, b]\), then \(\displaystyle \int_{a}^{b} f(x) d x=F(b)-F(a)\) where \(F\) is any antiderivative of \(\mathrm{f}\), that is, a function such that \(\mathrm{F}^{\prime}=\mathrm{f}.\)

    Note : If \(\displaystyle \int_{a}^{b} f(x) d x=0 \Rightarrow\) then the equation \(f(x)=0\) has atleast one root lying in \((a, b)\) provided \(f\) is a continuous function in \((a, b)\).


    2. Representation of Definite Integration

     A definite integral is denoted by \(\displaystyle \int_{a}^{b}  f(x) d x\) which represents the algebraic area bounded by the curve \(y=\mathrm{f}(\mathrm{x})\), the ordinates \(\mathrm{x}=\mathrm{a}\), \(\mathrm{x}=\mathrm{b}\) and the \(\mathrm{x}\) -axis. ex. \(\displaystyle \int_{0}^{2 \pi} \sin \mathrm{x} \mathrm{d} \mathrm{x}=0\)


    3. PROPERTIES OF DEFINITE INTEGRAL:

    (a) \(\displaystyle \int_{a}^{b} f(x) d x=\displaystyle \int_{a}^{b} f(t) d t \Rightarrow \displaystyle \int_{a}^{b} f(x) d x\) does not depend upon \(x\). It is a numerical quantity.

    (b) \(\displaystyle \int_{a}^{b} f(x) d x=-\displaystyle \int_{b}^{a} f(x) d x\)

    (c) \(\displaystyle \int_{a}^{b} f(x) d x=\displaystyle \int_{a}^{c} f(x) d x+\displaystyle \int_{c}^{b} f(x) d x\), where \(c\) may lie inside or outside the interval \([\mathrm{a}, \mathrm{b}]\). This property to be used when \(\mathrm{f}\) is piecewise continuous in \((a, b)\).

    (d) \(\displaystyle \int_{-a}^{a} f(x) d x\)\(=\displaystyle \int_{0}^{a}[f(x)+f(-x)] d x\)\(=\left[\begin{array}{ll}0  \,\, & \text { if } f(x) \text { is an odd function } \\ 2 \displaystyle \int_{0}^{a} f(x) \,\, d x  & \text { if } f(x) \text { is an even function }\end{array}\right.\)

    (e) \(\displaystyle \int_{a}^{b} f(x) d x=\displaystyle \int_{a}^{b} f(a+b-x) d x\), In particular \(\displaystyle \int_{0}^{a} f(x) d x=\displaystyle \int_{0}^{a} f(a-x) d x\)

    (f) \(\displaystyle \int_{0}^{2 a} f(x) d x\)\(=\displaystyle \int_{0}^{a} f(x) d x+\displaystyle \int_{0}^{a} f(2 a-x) d x\)\(=\left[\begin{array}{l}2 \displaystyle \int_{0}^{a}(x) d x  & \text { if } f(2 a-x)=f(x) \\ 0  & \text { if } f(2 a-x)=-f(x)\end{array}\right.\)

    (g) \(\displaystyle \int_{0}^{\mathrm{nT}} \mathrm{f}(\mathrm{x}) \mathrm{d} \mathrm{x}=\mathrm{n} \displaystyle \int_{0}^{\mathrm{T}} \mathrm{f}(\mathrm{x}) \mathrm{dx}, \quad(\mathrm{n} \in \mathrm{I}) ;\) where '\(\mathrm{T}\)' is the period of the function i.e. \(f(T+x)=f(x)\)

    Note that : \(\displaystyle \int_{x}^{T+x} f(t)\) dt will be independent of \(x\) and equal to \(\displaystyle \int_{0}^{T} f(t)\) dt

    (h) \(\displaystyle \int_{\mathrm{a}+\mathrm{nT}}^{\mathrm{b}+\mathrm{nT}} \mathrm{f}(\mathrm{x}) \mathrm{d} \mathrm{x}=\displaystyle \int_{\mathrm{a}}^{\mathrm{b}} \mathrm{f}(\mathrm{x}) \mathrm{dx}\) where \(\mathrm{f}(\mathrm{x})\) is periodic with period \(\mathrm{T} \) & \(\mathrm{n} \in \mathrm{I}\).

    (i) \(\displaystyle \int_{m a}^{n a} f(x) d x=(n-m) \displaystyle \int_{0}^{a} f(x) d x,(n, m \in D)\) if \(f(x)\) is periodic with period'a'.


    4. WALLI'S FORMULA :

    (a) \(\displaystyle \int_{0}^{\pi / 2} \sin ^{n} x d x\)\(=\displaystyle \int_{0}^{\pi / 2} \cos ^{n} x d x\)\(=\dfrac{(n-1)(n-3) \ldots .(1 \text { or } 2)}{n(n-2) \ldots . .(1 \text { or } 2)} K\)

    where \(K=\left\{\begin{array}{ll}\pi / 2 & \text { if } n \text { is even } \\ 1 & \text { if } n \text { is odd }\end{array}\right.\)

    (b) \(\displaystyle \int_{0}^{\pi / 2} \sin ^{n} x \cdot \cos ^{m} x d x\)

    \(\scriptsize{=\dfrac{[(\mathrm{n}-1)(\mathrm{n}-3)(\mathrm{n}-5) \ldots 1 \text { or } 2][(\mathrm{m}-1)(\mathrm{m}-3) \ldots .1 \text { or } 2]}{(\mathrm{m}+\mathrm{n})(\mathrm{m}+\mathrm{n}-2)(\mathrm{m}+\mathrm{n}-4) \ldots .1 \text { or } 2} \mathrm{~K}}\)

    Where \(\mathrm{K}=\left\{\begin{array}{ll}\dfrac{\pi}{2} & \text { if both } \mathrm{m} \text { and } \mathrm{n} \text { are even }(\mathrm{m}, \mathrm{n} \in \mathrm{N}) \\ 1 & \text { otherwise }\end{array}\right.\)


    5. DERIVATIVE OF ANTIDERIVATIVE FUNCTION (NewtonLeibnitz Formula)

    If \(\mathrm{h}(\mathrm{x}) \) & \(\mathrm{~g}(\mathrm{x})\) are differentiable functions of \(\mathrm{x}\) then

    \(\dfrac{\mathrm{d}}{\mathrm{d} \mathrm{x}} \displaystyle \int_{\mathrm{g}(\mathrm{x})}^{\mathrm{h}(\mathrm{x})} \mathrm{f}(\mathrm{t}) \mathrm{d} \mathrm{t}=\mathrm{f}[\mathrm{h}(\mathrm{x})] \cdot h^{\prime}(\mathrm{x})-\mathrm{f}[\mathrm{g}(\mathrm{x})] \cdot \cdot \mathrm{g}^{\prime}(\mathrm{x})\)


    6. DEFINITE INTEGRAL AS LIMIT OF A SUM 

    \(\begin{aligned} \displaystyle \int_{a}^{b}(x) d x &= \displaystyle \lim_{n \rightarrow \infty}h[f(a)+f(a+h)+f(a+2 h)+\ldots \\ & . .+f(a+\overline{n-1} h)] \\ &=\displaystyle \lim_{h \rightarrow \infty} h\sum_{r=0}^{n-1} f(a+r h), \text { where } b-a=n h \end{aligned}\)

    If \(a=0 \) & \(b=1\) then, \(\displaystyle \lim_{n \rightarrow \infty} \sum_{r=0}^{n-1} f(r h)=\displaystyle \int_{0}^{1} f(x) d x ;\) where \(n h=1\)

    OR \( \displaystyle \lim_{n \rightarrow \infty}\left(\dfrac{1}{n}\right) \sum_{r=1}^{n-1} f\left(\dfrac{r}{n}\right)=\displaystyle \int_{0}^{1} f(x) d x\)


    7. ESTIMATION OF DEFINTE INTEGRAL 

    (a) If \(\mathrm{f}(\mathrm{x})\) is continuous in \([\mathrm{a}, \mathrm{b}]\) and it's range in this interval is \([\mathrm{m},\) M], then \(m(b-a) \leq \displaystyle \int_{a}^{b} f(x) d x \leq M(b-a)\)

    (b) If \(\mathrm{f}(\mathrm{x}) \leq \phi(\mathrm{x})\) for \(\mathrm{a} \leq \mathrm{x} \leq \mathrm{b}\) then \(\displaystyle \int_{\mathrm{a}}^{\mathrm{b}} \mathrm{f}(\mathrm{x}) \mathrm{dx} \leq \displaystyle \int_{\mathrm{a}}^{\mathrm{b}} \phi(\mathrm{x}) \mathrm{dx}\)

    (c) \(\left|\displaystyle \int_{a}^{b} f(x) d x\right| \leq \displaystyle \int_{a}^{b}|f(x)| d x\).

    (d) If \(f(x) \geq 0\) on the interval \([a, b]\), then \(\displaystyle \int_{a}^{b} f(x) d x \geq 0\).

    (e) \(\mathrm{f}(\mathrm{x})\) and \(\mathrm{g}(\mathrm{x})\) are two continuous function on \([\mathrm{a}, \mathrm{b}]\) then \(\left|\displaystyle \int_{a}^{b} f(x) g(x) d x\right| \leq \sqrt{\displaystyle \int_{a}^{b} f^{2}(x) d x \displaystyle \int_{a}^{b} g^{2}(x) d x}\)


    8. SOME STANDARD RESULTS :

    (a) \(\displaystyle \int_{0}^{\pi / 2} \log \sin x d x=-\dfrac{\pi}{2} \log 2=\displaystyle \int_{0}^{\pi / 2} \log \cos x d x\)

    (b) \(\displaystyle \int_{a}^{b}\{x\} d x=\dfrac{b-a}{2} ; a, b \in I\)

    (c) \(\displaystyle \int_{a}^{b} \dfrac{|x|}{x} d x=|b|-|a|\).


    Labels:

    Comments

    Post a Comment

    Popular posts from this blog

    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}{...
    Post a Comment
    Read more

    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{...
    Post a Comment
    Read more

    Hyperbola - Notes, Concept and All Important Formula

    HYPERBOLA The Hyperbola is a conic whose eccentricity is greater than unity \((e>1) .\) 1. STANDARD EQUATION & DEFINITION(S): Standard equation of the hyperbola is \(\dfrac{\mathbf{x}^{2}}{\mathbf{a}^{2}}-\dfrac{\mathbf{y}^{2}}{\mathbf{b}^{2}}=\mathbf{1},\) where \(b^{2}=a^{2}\left(e^{2}-1\right)\) or \(a^{2} e^{2}=a^{2}+b^{2}\)    i.e.    \(e^{2}=1+\dfrac{b^{2}}{a^{2}}\) \(=1+\left(\dfrac{\text { Conjugate Axis }}{\text { Transverse Axis }}\right)^{2}\) (a) Foci : \(\mathrm{S} \equiv(\mathrm{a} e, 0) \quad \& \quad \mathrm{~S}^{\prime} \equiv(-\mathrm{a} e, 0) .\) (b) Equations of directrices: \(\mathrm{x}=\dfrac{\mathrm{a}}{e}\quad \) & \(\quad \mathrm{x}=-\dfrac{\mathrm{a}}{e}\) (c) Vertices: \(A \equiv(a, 0)\quad \) & \(\quad A^{\prime} \equiv(-a, 0)\) (d) Latus rectum: (i) Equation: \(\mathrm{x}=\pm \mathrm{ae}\) (ii) Length:  \(\begin{aligned} &=\dfrac{2 b^{2}}{a}=\dfrac{(\text { Conjugate Axis })^{2}}{(\text { Transverse Axis ...
    Post a Comment
    Read more