We have
$I=\int\limits_{0}^{1}\left(1-x^{50}\right)\left(1-x^{50}\right)^{100} d x $
$=\int\limits_{0}^{1}\left(1-x^{50}\right)^{100} d x-\int\limits_{0}^{1} x x^{49}\left(1-x^{50}\right)^{100} d x$
Therefore,
$I_{1}=\int\limits_{0}^{1} x x^{49}\left(1-x^{50}\right)^{100} dx$
$1-x^{50}= t \Rightarrow -50 x^{49} dx = dt$
$\Rightarrow x^{49} dx =\frac{-d t}{50}$
$=\left.\left(-\frac{x}{50}\right) \frac{\left(1-x^{50}\right)^{101}}{101}\right|_{0} ^{1}+\frac{1}{5050} \int\limits_{0}^{1} \frac{\left(1-x^{50}\right)^{101}}{1} dx$
Now, $I=-\frac{1}{5050} I+\int\limits_{0}^{1}\left(1-x^{50}\right)^{100} d x$
$\left(\frac{5051}{5050}\right) \int\limits_{0}^{1}\left(1-x^{50}\right)^{101} d x=\int\limits_{0}^{1}\left(1-x^{50}\right)^{100} d x$
$5051=5050 \frac{\int\limits_{0}^{1}\left(1-x^{50}\right)^{100} d x}{1} \int\limits_{0}^{50}\left(1-x^{101} d x\right.$