Answer
414.9k+ views
Hint: We have to determine whether the antiderivative of every odd function is odd or even function. We will use the definition of odd function which is given as-
A function is odd if and only if ${\text{f}}\left( {{\text{ - x}}} \right) = - {\text{f}}\left( {\text{x}} \right)$ . Let us assume the function to be x. Then integrate the given function to find the antiderivative and name it g(x). Then check whether this function is odd or even if it is odd then ${\text{f}}\left( {{\text{ - x}}} \right) = - {\text{f}}\left( {\text{x}} \right)$ and if it is even then ${\text{f}}\left( {{\text{ - x}}} \right) = {\text{f}}\left( {\text{x}} \right)$.
Complete step-by-step answer:
We have to find the anti-derivative of every odd function.
Let us assume that the function be ${\text{f}}\left( {\text{x}} \right){\text{ = x}}$ which represents an odd function. We know that a function is odd if and only if ${\text{f}}\left( {{\text{ - x}}} \right) = - {\text{f}}\left( {\text{x}} \right)$.
So suppose if x=$2$ then according to definition of odd function-
$ \Rightarrow f\left( { - 2} \right) = - f\left( 2 \right)$
So we can write-
$ \Rightarrow - 2 = - 2$
So f(x) =x is an odd function it is verified.
Now we have to find the anti-derivative of the given function. So to find the anti-derivative of the function, we will integrate the given function-
$ \Rightarrow \int {{\text{f}}\left( {\text{x}} \right)} dx = \int x dx$
Now we know that formula of integration of x is given as-
$ \Rightarrow \int {{x^n} = \dfrac{{{x^{n + 1}}}}{{n + 1}}} $
On using this formula we get-
$ \Rightarrow \int {{\text{f}}\left( {\text{x}} \right)} dx = \dfrac{{{x^{1 + 1}}}}{{1 + 1}} + C$ ,Where C is integration constant.
On solving, we get-
$ \Rightarrow \int {{\text{f}}\left( {\text{x}} \right)} dx = \dfrac{{{x^2}}}{2} + C$
Now we can write the anti-derivative of f(x) as g(x).
Then we can write-
$ \Rightarrow {\text{g}}\left( {\text{x}} \right) = \dfrac{{{x^2}}}{2} + C$
$ \Rightarrow {\text{g}}\left( {{\text{ - x}}} \right) = \dfrac{{{{\left( { - x} \right)}^2}}}{2} + C$
Now we know that square of any function will always give us an even function
$ \Rightarrow {\text{g}}\left( {{\text{ - x}}} \right) = \dfrac{{{x^2}}}{2} + C$
So it is an even function as an even function is defined as-
A function is even if and only if ${\text{f}}\left( {{\text{ - x}}} \right){\text{ = f}}\left( {\text{x}} \right)$
We can also verify it by putting any number in place of x like-
$ \Rightarrow {\text{g}}\left( {{\text{ - 2}}} \right) = \dfrac{{{{\left( { - 2} \right)}^2}}}{2} = \dfrac{4}{2}$
So it is proved that the anti-derivative of an odd function is an even function.
The correct answer is B.
Note: Here we integrate the given function because integral of a function is equal to the anti-derivative of the function. The even and odd functions tell us about the symmetry of the function along x-axis or y-axis.
If a function is even then the graph of that function will be symmetric about the y-axis.
If a function is an odd function then the graph of that function will be symmetric about the x-axis.
A function is odd if and only if ${\text{f}}\left( {{\text{ - x}}} \right) = - {\text{f}}\left( {\text{x}} \right)$ . Let us assume the function to be x. Then integrate the given function to find the antiderivative and name it g(x). Then check whether this function is odd or even if it is odd then ${\text{f}}\left( {{\text{ - x}}} \right) = - {\text{f}}\left( {\text{x}} \right)$ and if it is even then ${\text{f}}\left( {{\text{ - x}}} \right) = {\text{f}}\left( {\text{x}} \right)$.
Complete step-by-step answer:
We have to find the anti-derivative of every odd function.
Let us assume that the function be ${\text{f}}\left( {\text{x}} \right){\text{ = x}}$ which represents an odd function. We know that a function is odd if and only if ${\text{f}}\left( {{\text{ - x}}} \right) = - {\text{f}}\left( {\text{x}} \right)$.
So suppose if x=$2$ then according to definition of odd function-
$ \Rightarrow f\left( { - 2} \right) = - f\left( 2 \right)$
So we can write-
$ \Rightarrow - 2 = - 2$
So f(x) =x is an odd function it is verified.
Now we have to find the anti-derivative of the given function. So to find the anti-derivative of the function, we will integrate the given function-
$ \Rightarrow \int {{\text{f}}\left( {\text{x}} \right)} dx = \int x dx$
Now we know that formula of integration of x is given as-
$ \Rightarrow \int {{x^n} = \dfrac{{{x^{n + 1}}}}{{n + 1}}} $
On using this formula we get-
$ \Rightarrow \int {{\text{f}}\left( {\text{x}} \right)} dx = \dfrac{{{x^{1 + 1}}}}{{1 + 1}} + C$ ,Where C is integration constant.
On solving, we get-
$ \Rightarrow \int {{\text{f}}\left( {\text{x}} \right)} dx = \dfrac{{{x^2}}}{2} + C$
Now we can write the anti-derivative of f(x) as g(x).
Then we can write-
$ \Rightarrow {\text{g}}\left( {\text{x}} \right) = \dfrac{{{x^2}}}{2} + C$
$ \Rightarrow {\text{g}}\left( {{\text{ - x}}} \right) = \dfrac{{{{\left( { - x} \right)}^2}}}{2} + C$
Now we know that square of any function will always give us an even function
$ \Rightarrow {\text{g}}\left( {{\text{ - x}}} \right) = \dfrac{{{x^2}}}{2} + C$
So it is an even function as an even function is defined as-
A function is even if and only if ${\text{f}}\left( {{\text{ - x}}} \right){\text{ = f}}\left( {\text{x}} \right)$
We can also verify it by putting any number in place of x like-
$ \Rightarrow {\text{g}}\left( {{\text{ - 2}}} \right) = \dfrac{{{{\left( { - 2} \right)}^2}}}{2} = \dfrac{4}{2}$
So it is proved that the anti-derivative of an odd function is an even function.
The correct answer is B.
Note: Here we integrate the given function because integral of a function is equal to the anti-derivative of the function. The even and odd functions tell us about the symmetry of the function along x-axis or y-axis.
If a function is even then the graph of that function will be symmetric about the y-axis.
If a function is an odd function then the graph of that function will be symmetric about the x-axis.
Recently Updated Pages
How many sigma and pi bonds are present in HCequiv class 11 chemistry CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Why Are Noble Gases NonReactive class 11 chemistry CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Let X and Y be the sets of all positive divisors of class 11 maths CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Let x and y be 2 real numbers which satisfy the equations class 11 maths CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Let x 4log 2sqrt 9k 1 + 7 and y dfrac132log 2sqrt5 class 11 maths CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Let x22ax+b20 and x22bx+a20 be two equations Then the class 11 maths CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Trending doubts
Fill the blanks with the suitable prepositions 1 The class 9 english CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
At which age domestication of animals started A Neolithic class 11 social science CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Which are the Top 10 Largest Countries of the World?
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Give 10 examples for herbs , shrubs , climbers , creepers
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Difference Between Plant Cell and Animal Cell
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Write a letter to the principal requesting him to grant class 10 english CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Change the following sentences into negative and interrogative class 10 english CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Fill in the blanks A 1 lakh ten thousand B 1 million class 9 maths CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)