Test Index

NCERT Class XII Mathematics Chapter - - Solutions

Show Para  Hide Para 
Exercise - 7.11
By using the properties of definite integrals, evaluate the integrals in Exercises 208 to 225.
© examsnet.com
Question : 213 of 271
Marks: +1, -0
∫01x(1−x)ndx\int\limits_{0}^{1}x(1-x)^ndx
Solution:  
Let I=∫01x(1−x)ndxI=\int\limits_{0}^{1}x(1-x)^ndx
⇒I=∫01(1−x)[1−(1−x)]ndx\Rightarrow I=\int\limits_{0}^{1}(1-x)[1-(1-x)]^ndx [∵∫0af(x)dx=∫0af(a−x)dx]\left[\because \int\limits_{0}^{a}f(x)dx=\int\limits_{0}^{a}f(a-x)dx\right]
=∫01(1−x)xndx=\int\limits_{0}^{1}(1-x)x^ndx
=∫01(xn−xn+1)dx=\int\limits_{0}^{1}(x^n-x^{n+1})dx
=[xn+1n+1−xn+2n+2]01=\left[\frac{x^{n+1}}{n+1}-\frac{x^{n+2}}{n+2}\right]_{0}^{1}
=[1n+1−1n+2]−(0−0)=\left[\frac{1}{n+1}-\frac{1}{n+2}\right]-(0-0)
=1(n+1)(n+2)=\frac{1}{(n+1)(n+2)}
© examsnet.com
Go to Question:
Prev Question
Next Question
Advertisements