Answer
452.7k+ views
Hint: Here, we will be proceeding by using the property of the definite integral which is $\int_a^b {\left[ {f(x)} \right]} dx = \int_a^b {\left[ {f(a + b - x)} \right]} dx$ where $f(x)$ is any function of x.
Complete step-by-step answer:
Let the given integral be ${\text{I}} = \int_0^{10} {\left[ {\dfrac{{{x^{10}}}}{{{{\left( {10 - x} \right)}^{10}} + {x^{10}}}}} \right]} dx{\text{ }} \to {\text{(1)}}$
According to the property of definite integral, we have
$\int_a^b {\left[ {f(x)} \right]} dx = \int_a^b {\left[ {f(a + b - x)} \right]} dx$
Using the above property, the integral given in equation (1) becomes
\[
{\text{I}} = \int_0^{10} {\left[ {\dfrac{{{{\left( {10 + 0 - x} \right)}^{10}}}}{{{{\left[ {10 - \left( {10 + 0 - x} \right)} \right]}^{10}} + {{\left( {10 + 0 - x} \right)}^{10}}}}} \right]} dx = \int_0^{10} {\left[ {\dfrac{{{{\left( {10 - x} \right)}^{10}}}}{{{{\left[ {10 - 10 + x} \right]}^{10}} + {{\left( {10 - x} \right)}^{10}}}}} \right]} dx \\
{\text{I}} = \int_0^{10} {\left[ {\dfrac{{{{\left( {10 - x} \right)}^{10}}}}{{{x^{10}} + {{\left( {10 - x} \right)}^{10}}}}} \right]} dx{\text{ }} \to {\text{(2)}} \\
\]
By adding equations (1) and (2), we get
$
{\text{I}} + {\text{I}} = \int_0^{10} {\left[ {\dfrac{{{x^{10}}}}{{{{\left( {10 - x} \right)}^{10}} + {x^{10}}}}} \right]} dx + \int_0^{10} {\left[ {\dfrac{{{{\left( {10 - x} \right)}^{10}}}}{{{x^{10}} + {{\left( {10 - x} \right)}^{10}}}}} \right]} dx \\
\Rightarrow 2{\text{I}} = \int_0^{10} {\left[ {\dfrac{{{x^{10}}}}{{{{\left( {10 - x} \right)}^{10}} + {x^{10}}}} + \dfrac{{{{\left( {10 - x} \right)}^{10}}}}{{{x^{10}} + {{\left( {10 - x} \right)}^{10}}}}} \right]} dx = \int_0^{10} {\left[ {\dfrac{{{x^{10}} + {{\left( {10 - x} \right)}^{10}}}}{{{{\left( {10 - x} \right)}^{10}} + {x^{10}}}}} \right]} dx \\
\Rightarrow 2{\text{I}} = \int_0^{10} {\left( 1 \right)} dx = \left[ x \right]_0^{10} = \left[ {10 - 0} \right] = 10 \\
\Rightarrow {\text{I}} = 5 \\
$
So, the value of the integral $\int_0^{10} {\left[ {\dfrac{{{x^{10}}}}{{{{\left( {10 - x} \right)}^{10}} + {x^{10}}}}} \right]} dx$ is 5.
Note: In these type of problems, we somehow convert the complex function given in terms of x which is inside the integral (here it is $\dfrac{{{x^{10}}}}{{{{\left( {10 - x} \right)}^{10}} + {x^{10}}}}$) into a simpler function (here it comes out to be 1) using some property of the definite integral so that the integral of the function can be easily evaluated.
Complete step-by-step answer:
Let the given integral be ${\text{I}} = \int_0^{10} {\left[ {\dfrac{{{x^{10}}}}{{{{\left( {10 - x} \right)}^{10}} + {x^{10}}}}} \right]} dx{\text{ }} \to {\text{(1)}}$
According to the property of definite integral, we have
$\int_a^b {\left[ {f(x)} \right]} dx = \int_a^b {\left[ {f(a + b - x)} \right]} dx$
Using the above property, the integral given in equation (1) becomes
\[
{\text{I}} = \int_0^{10} {\left[ {\dfrac{{{{\left( {10 + 0 - x} \right)}^{10}}}}{{{{\left[ {10 - \left( {10 + 0 - x} \right)} \right]}^{10}} + {{\left( {10 + 0 - x} \right)}^{10}}}}} \right]} dx = \int_0^{10} {\left[ {\dfrac{{{{\left( {10 - x} \right)}^{10}}}}{{{{\left[ {10 - 10 + x} \right]}^{10}} + {{\left( {10 - x} \right)}^{10}}}}} \right]} dx \\
{\text{I}} = \int_0^{10} {\left[ {\dfrac{{{{\left( {10 - x} \right)}^{10}}}}{{{x^{10}} + {{\left( {10 - x} \right)}^{10}}}}} \right]} dx{\text{ }} \to {\text{(2)}} \\
\]
By adding equations (1) and (2), we get
$
{\text{I}} + {\text{I}} = \int_0^{10} {\left[ {\dfrac{{{x^{10}}}}{{{{\left( {10 - x} \right)}^{10}} + {x^{10}}}}} \right]} dx + \int_0^{10} {\left[ {\dfrac{{{{\left( {10 - x} \right)}^{10}}}}{{{x^{10}} + {{\left( {10 - x} \right)}^{10}}}}} \right]} dx \\
\Rightarrow 2{\text{I}} = \int_0^{10} {\left[ {\dfrac{{{x^{10}}}}{{{{\left( {10 - x} \right)}^{10}} + {x^{10}}}} + \dfrac{{{{\left( {10 - x} \right)}^{10}}}}{{{x^{10}} + {{\left( {10 - x} \right)}^{10}}}}} \right]} dx = \int_0^{10} {\left[ {\dfrac{{{x^{10}} + {{\left( {10 - x} \right)}^{10}}}}{{{{\left( {10 - x} \right)}^{10}} + {x^{10}}}}} \right]} dx \\
\Rightarrow 2{\text{I}} = \int_0^{10} {\left( 1 \right)} dx = \left[ x \right]_0^{10} = \left[ {10 - 0} \right] = 10 \\
\Rightarrow {\text{I}} = 5 \\
$
So, the value of the integral $\int_0^{10} {\left[ {\dfrac{{{x^{10}}}}{{{{\left( {10 - x} \right)}^{10}} + {x^{10}}}}} \right]} dx$ is 5.
Note: In these type of problems, we somehow convert the complex function given in terms of x which is inside the integral (here it is $\dfrac{{{x^{10}}}}{{{{\left( {10 - x} \right)}^{10}} + {x^{10}}}}$) into a simpler function (here it comes out to be 1) using some property of the definite integral so that the integral of the function can be easily evaluated.
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)