   Question Answers

# If function f(x) is an odd function, then find the value of following integral:$\int_{-a}^{a}{f(x)dx}$A) $2\int_{0}^{a}{f(x)dx}$B) $\int_{0}^{a}{f(x)dx}$C) 0D) $\int_{0}^{a}{f(a-x)dx}$  Hint: Integrate the given expression directly and use the concept given by ‘the integration of an Odd function is always even.’

We will write the expression, of which we have to find the value and assume it as L,
$L=\int_{-a}^{a}{f(x)dx}$……………………………… (1)
Suppose,
$\int{f(x)}dx=F(x)$…………………………………. (2)
To proceed further we should know the formula of definite integration given below,
Formula:
$\int\limits_{a}^{b}{f(x)dx}=\left[ F(x) \right]_{a}^{b}$
By using the formula and equation (2) we can write equation (1) as,
$\therefore L=\left[ F(x) \right]_{-a}^{a}$
To proceed further we should know how to substitute the limits in f(x) which is shown below,
$\left[ F(x) \right]_{a}^{b}=F(b)-F(a)$
By using above formula we can write ‘L’ as shown below,
$\therefore L=\left[ F(x) \right]_{-a}^{a}=F(a)-F(-a)$
$\therefore L=F(a)-F(-a)$………………………………………… (3)
To solve the above equation further we should know the concept of integration of odd functions given below,
Concept:
Integration of an Odd function is always even.
As F(x) is the integration of f(x) therefore it is an even function and therefore we should know how the even functions are expressed.
As F(x) is an even function therefore we should know the concept given below to proceed further in the solution,
Concept:
If F(x) is an even function then,
F(-x) = F(x), For all values of x.
By using the above concept we can write the equation (3) as,
$\therefore L=F(a)-F(a)$
$\therefore L=0$………………………………………. (4)
Therefore from equation (4) and equation (1) we can write,
$\therefore L=\int_{-a}^{a}{f(x)dx}=0$
Therefore if f(x) is an odd function then $\int_{-a}^{a}{f(x)dx}$ is equal to ‘0’
Therefore option (c) is the correct answer.

Note: You should know that, if F(x) is an even function then, F(-x) = F(x) to get the right answer.
View Notes
Analytic Function  One to One Function  Kidney Function Test  The Difference Between an Animal that is A Regulator and One that is A Conformer  What Happens if the Earth Stops Rotating?  CBSE Class 11 Maths Formulas  What is the Difference Between Trade and Commerce?  What is the Cell Envelope?  What is the Placebo Effect  What is the full form of phd?  Important Questions for CBSE Class 11 Maths Chapter 2 - Relations and Functions  Important Questions for CBSE Class 7 English An Alien Hand Chapter 4 - The Cop and The Anthem  Important Questions for CBSE Class 7 English An Alien Hand Chapter 1 - The Tiny Teacher  Important Questions for CBSE Class 7 English An Alien Hand Chapter 3 - The Desert  Important Questions for CBSE Class 11 Maths Chapter 3 - Trigonometric Functions  Important Questions for CBSE Class 7 English An Alien Hand  Important Questions for CBSE Class 7 English An Alien Hand Chapter 8 - The Bear Story  Important Questions for CBSE Class 11 Chemistry Chapter 11 - The p-Block Elements  Important Questions for CBSE Class 11 Maths  Important Questions for CBSE Class 11 English Woven Chapter 6 - The Story  CBSE Class 12 Maths Question Paper 2020  CBSE Class 10 Maths Question Paper 2020  Maths Question Paper for CBSE Class 10 - 2011  Maths Question Paper for CBSE Class 10 - 2008  CBSE Class 10 Maths Question Paper 2017  Maths Question Paper for CBSE Class 10 - 2012  Maths Question Paper for CBSE Class 10 - 2009  Maths Question Paper for CBSE Class 10 - 2010  Maths Question Paper for CBSE Class 10 - 2007  Maths Question Paper for CBSE Class 12 - 2013  Relations and Functions NCERT Solutions - Class 11 Maths  NCERT Solutions for Class 7 English An Alien Hand - The Cop And The Anthem  NCERT Solutions for Class 7 English An Alien Hand - The Desert  NCERT Solutions for Class 9 English Beehive Chapter 11 - If I Were You  NCERT Solutions for Class 11 Maths Chapter 3  NCERT Solutions Class 11 English Woven Words Poem Chapter 5 The World is too Much With Us  RD Sharma Class 11 Maths Solutions Chapter 23 - The Straight Lines  RD Sharma Class 11 Maths Solutions Chapter 24 - The Circle  RD Sharma Class 11 Maths Solutions Chapter 3 - Functions  NCERT Solutions for Class 11 Economics - Chapter 3 - Liberalisation, Privatisation and Globalisation: An Appraisal  