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

    Determinant - Notes, Concept and All Important Formula

    DETERMINANT 1. MINORS : The minor of a given element of determinant is the determinant of the elements which remain after deleting the row & the column in which the given element stands. For example, the minor of \(a _{1}\) in \(\left|\begin{array}{lll} a _{1} & b _{1} & c _{1} \\ a _{2} & b _{2} & c _{2} \\ a _{3} & b _{3} & c _{3}\end{array}\right|\) is \(\left|\begin{array}{ll} b _{2} & c _{2} \\ b _{3} & c _{3}\end{array}\right| \&\) the minor of \(b_{2}\) is \(\quad\left|\begin{array}{ll}a_{1} & c_{1} \\ a_{3} & c_{3}\end{array}\right|\) . Hence a determinant of order three will have " 9 minors". 2. COFACTORS : If \(M _{ ij }\) represents the minor of the element belonging to \(i ^{\text {th }}\) row and \(j ^{\text {th }}\) column then the cofactor of that element : \(C_{i j}=(-1)^{i+j} \cdot M_{i j}\) Important Note : Consider \(\Delta=\left|\begin{array}{lll}a_{1} & a_{2} & a_{3} \\ b_{1} & b_{2} ...
    Post a Comment
    Read more

    Circle- Notes, Concept and All Important Formula

    CIRCLE 1. DEFINITION : A circle is the locus of a point which moves in a plane in such a way that its distance from a fixed point remains constant. The fixed point is called the centre of the circle and the constant distance is called the radius of the circle. All Chapter Notes, Concept and Important Formula 2. STANDARD EQUATIONS OF THE CIRCLE : (a) Central Form: If \((\mathrm{h}, \mathrm{k})\) is the centre and \(\mathrm{r}\) is the radius of the circle then its equation is \((\mathbf{x}-\mathbf{h})^{2}+(\mathbf{y}-\mathbf{k})^{2}=\mathbf{r}^{2}\) (b) General equation of circle : \(\mathbf{x}^{2}+\mathbf{y}^{2}+\mathbf{2 g x}+\mathbf{2 f y}+\mathbf{c}=\mathbf{0}\) , where \(g, \mathrm{f}, c\) are constants and centre is \((-g,-f)\) i.e. \(\left(-\frac{\text { coefficient of } \mathrm{x}}{2},-\frac{\text { coefficient of } \mathrm{y}}{2}\right)\) and radius \(r=\sqrt{g^{2}+f^{2}-c}\) Note : The general quadratic equation in \(\mathrm{x}\) and \(\mathrm{y}\) , \(a x^{2}+b y^{2}+2 ...
    Post a Comment
    Read more

    Difference and relation between Differentiation and Integration

    Relation between Differentiation and Integration Table Of Contents Look at the information given below. \[\mathbf{ y=f(x)}\] \[ \mathbf{ f'(x)\rightarrow \text{Derivatives of f(x)}}\] \[ \mathbf{\displaystyle \int_a^b f’(x) = ?}\] Can you tell me the value of above integral? Yes, it will be equal to f(b)-f(a) . We have already known this result. It tells us that integration is just the reverse of differentiation, integral of the derivative of the function f(x) is just equal to the difference in the function f(x) evaluated at the limits of integration. Indefinite integration- Notes and Formula Part 1 Now with this topic, we will understand how to apply this result to find the integral of a function.  Consider this function \(\mathbf{g(x)=x^2}\) Let's find the integral of this function from a to b i.e \(\mathbf{\displaystyle \int_a^b g(x) \, dx}\) .  Can you think how we can apply this result to find ...
    Post a Comment
    Read more