Question

The value of $\frac{(5050)\underset{0}{\overset{1}{\int }}{\left(1-x^{50}\right)}^{100}dx}{\underset{0}{\overset{1}{\int }}{\left(1-x^{50}\right)}^{101}dx}$ is ...

Answer 1

Answer: 5051

We have
$I=\underset{0}{\overset{1}{\int }}\left(1-x^{50}\right){\left(1-x^{50}\right)}^{100}dx$
$=\underset{0}{\overset{1}{\int }}{\left(1-x^{50}\right)}^{100}dx-\underset{0}{\overset{1}{\int }}x{x}^{49}{\left(1-x^{50}\right)}^{100}dx$
Therefore,
$I_1=\underset{0}{\overset{1}{\int }}x{x}^{49}{\left(1-x^{50}\right)}^{100}dx$
$1-x^{50}=t\Rightarrow -50x^{49}dx=dt$
$\Rightarrow x^{49}dx=\frac{-dt}{50}$
$={\left(-\frac{x}{50}\right)\frac{{\left(1-x^{50}\right)}^{101}}{101}|}_0^1+\frac{1}{5050}\underset{0}{\overset{1}{\int }}\frac{{\left(1-x^{50}\right)}^{101}}{1}dx$
Now, $I=-\frac{1}{5050}I+\underset{0}{\overset{1}{\int }}{\left(1-x^{50}\right)}^{100}dx$
$\left(\frac{5051}{5050}\right)\underset{0}{\overset{1}{\int }}{\left(1-x^{50}\right)}^{101}dx=\underset{0}{\overset{1}{\int }}{\left(1-x^{50}\right)}^{100}dx$
$5051=5050\frac{\underset{0}{\overset{1}{\int }}{\left(1-x^{50}\right)}^{100}dx}{1}\underset{0}{\overset{50}{\int }}(1-x^{101}dx$