Definite Integration - Notes, Concept and All Important Formula

DEFINITE INTEGRATION

• 1. (a) The Fundamental Theorem of Calculus, Part 1:
• (b) The Fundamental Theorem of Calculus, Part 2:
• 2. Representation of Definite Integration
• 3. PROPERTIES OF DEFINITE INTEGRAL:
• 4. WALLI'S FORMULA :
• 5. DERIVATIVE OF ANTIDERIVATIVE FUNCTION (NewtonLeibnitz Formula)
• 6. DEFINITE INTEGRAL AS LIMIT OF A SUM 
• 7. ESTIMATION OF DEFINTE INTEGRAL 
• 8. SOME STANDARD RESULTS :

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 $(a, b)$.

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All Chapter Notes, Concept and Important Formula

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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. $\displaystyle \int_{0}^{2 \pi} \sin \mathrm{x} \mathrm{d} \mathrm{x}=0$

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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'.

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

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5. DERIVATIVE OF ANTIDERIVATIVE FUNCTION (NewtonLeibnitz Formula)

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

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

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

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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 $\left|\displaystyle \int_{a}^{b} f(x) g(x) d x\right| \leq \sqrt{\displaystyle \int_{a}^{b} f^{2}(x) d x \displaystyle \int_{a}^{b} g^{2}(x) d x}$

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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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