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  4. Application of Even- Odd Properties in Definite Integration - Practice Questions & MCQ

Application of Even- Odd Properties in Definite Integration - Practice Questions & MCQ

Updated on Sep 18, 2023 18:34 AM

Quick Facts

  • Application of Even- Odd Properties in Definite Integration is considered one the most difficult concept.

  • Application of Periodic Properties in Definite Integration is considered one of the most asked concept.

  • 36 Questions around this concept.

Concepts Covered - 2

Application of Even- Odd Properties in Definite Integration

Property 7

\\\mathrm{\;\;\;\;\;\;\;\;\;\;\;\;}\int_{-a}^{a}f(x)\;dx=\left\{\begin{matrix} 0, & \text { if } f \text { is an odd function }&\text{i.e. }f(-x)=-f(x)\\ \\ 2\int_{0}^{a}f(x) \;dx,&\text { if } f \text { is an even function } &\text{i.e. }f(-x)=f(x) \end{matrix}\right.

Proof: 

\\\mathrm{\int_{-a}^{a}f(x)\;dx=\underbrace{\int_{-a}^{0}f(x)\;dx}+\int_{0}^{a}f(x)\;dx}\\\text{\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;} x=-t\\\mathrm{\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;}=\int_{a}^{0}f(-t)\;(-dt)+\int_{0}^{a}f(x)\;dx\\\\\mathrm{\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;}=\int_{0}^{a}f(-x)\;(dx)+\int_{0}^{a}f(x)\;dx\\\\\mathrm{\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;}=\left\{\begin{matrix} -\int_{0}^{a}f(x)\;dx+\int_{0}^{a}f(x)\;dx, &\text{if f(x) is odd} \\ \\ \int_{0}^{a}f(x)\;dx+\int_{0}^{a}f(x)\;dx,& \text{if f(x) is even} \end{matrix}\right.\\\\\\\mathrm{\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;}=\left\{\begin{matrix} 0, & \text { if } f \text { is an odd function }\\ \\ 2\int_{0}^{a}f(x) \;dx,&\text { if } f \text { is an even function } \end{matrix}\right.

 

Proof using Graph

The graph of the odd function is symmetric about the origin, as shown in the above figure

\\\mathrm{So,\;\;if\;\;\int_{0}^{a}f(x)\;dx=\alpha\;\;then,\;\;\int_{-a}^{0}f(x)\;dx=-\alpha}\\\\\\\mathrm{\therefore \;\;\;\;\;\;\;\;\;\int_{-a}^{a}f(x)\;dx=0}

 

The graph of the even function is symmetric about the y-axis, as shown in the above figure

\\\mathrm{So,\;\;\;\;\;\int_{-a}^{0}f(x)\;dx=\;\;\int_{0}^{a}f(x)\;dx=\alpha}\\\\\\\mathrm{\therefore \;\;\;\;\;\;\;\;\;\int_{-a}^{a}f(x)\;dx=2\alpha=2\int_{0}^{a}f(x)\;dx}

 

Corollary:

\int_{0}^{2 a} f(x) d x=\left\{\begin{matrix} 2 \int_{0}^{a} f(x) d x, &&\text { if } f(2 a-x)=f(x)\\ \\ 0,&& \text { if } f(2 a-x)=-f(x) \end{matrix}\right.

Application of Periodic Properties in Definite Integration

Property 9

If f(x) is a periodic function with period T, then the area under f(x) for n periods would be n times the area under f(x) for one period, i.e.

\mathbf{\int_{0}^{n T} f(x) d x=n \int_{0}^{T} f(x) d x}

Proof:

Graphical Method

f(x) is a periodic function with period T. Consider the following graph of function f(x).

The graph of the function is the same in each of the interval (0, T), (T, 2T), (2T, 3T) ……..

So,

\\\mathrm{\int_{0}^{n T} f(x)\; d x=total\;shaded\;area}\\\mathrm{\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;=n\times\text{(area in the interval (0, T)})}\\\mathrm{\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;=n\int_{0}^{ T} f(x)\; d x}

 

Property 10


\\\mathbf{\int_{a}^{a+n T} f(x) d x=\int_{0}^{a T} f(x) d x=n \int_{0}^{T} f(x) d x}

Proof:

\\ \mathrm{Let,\;\;\;\;I=\int_{a}^{a+n T} f(x) \;d x}\\\\\mathrm{\;\;\;\;\;\;\;\;\;\;\;\;\;=\int_{a}^{0} f(x)\; d x+\int_{0}^{nT} f(x)\; d x+\underbrace{\int_{nT}^{a+nT} f(x)\; d x}}\\\text{\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;}x=y+nT\\\mathrm{\Rightarrow\;\;dx=dy\;\;and\;\;when\;x=nT\;\;then\;y=0\;\;and\;x=a+nT,\;y=a}\\\\\mathrm{\;\;\;\;\;\;\;\;\;\;\;\;\;=\int_{a}^{0} f(x)\; d x+\int_{0}^{nT} f(x)\; d x+\int_{0}^{a} f(y)\; d y}\\\\\mathrm{\;\;\;\;\;\;\;\;\;\;\;\;\;=n\int_{0}^{nT} f(x)\; d x}\\\\\mathrm{\left [ \because \int_{0}^{a} f(y)\; d y=\int_{0}^{a} f(x)\; d x\;\;\;and\;\;\int_{0}^{a} f(x)\; d x=-\int_{a}^{0} f(x)\; d x \right ]}

 

Property 11

\\\mathbf{\int_{a+n T}^{b+n T} f(x) d x=\int_{a}^{b} f(x) d x}

 

Property 12

\\\mathbf{\int_{m T}^{nT} f(x) \;d x=(n-m)\int_{0}^{T} f(x) d x}

Where ‘T’ is the period and m and n are Integers.

Study it with Videos

Application of Even- Odd Properties in Definite Integration
Application of Periodic Properties in Definite Integration

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Books

Reference Books

Application of Even- Odd Properties in Definite Integration

Mathematics for Joint Entrance Examination JEE (Advanced) : Calculus

Page No. : 8.22

Line : 23

Application of Periodic Properties in Definite Integration

Mathematics for Joint Entrance Examination JEE (Advanced) : Calculus

Page No. : 8.25

Line : 1

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