Question

The value of $\frac{(5050) \int \limits_0^1 (1-x^{50})^{100} dx} { \int \limits_0^1 (1-x^{50})^{100} dx } is$

Answer 1

Let $ I_2 = \int \limits_0^1 (1-x^{50 })^{101 } dx$
$=\left[\left(1-x^{50}\right)^{101} \cdot x\right]_{0}^{1}+\int\limits_{0}^{1}\left(1-x^{50}\right)^{100} 50 \cdot x^{49} \cdot x d x$
[using integration by parts]
$= 0- \int \limits_0^1 (50)(101)(1-x^{50})^{100}({-x}^{50}) dx $
$ = -(50)(101) \int \limits_0^1 (1-x^{50 })^{101} dx$
$ +(50)(101)\int \limits_0^1(1-x^{50})^{100} dx = 5050I_2 +5050 I_1$
$\therefore I_2+5050I_2 = 5050I_1 \Rightarrow \frac{(5050)I_2}{I_2 }=5051$