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


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