Question

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

Answer 1

Let $I_2=\underset{0}{\overset{1}{\int }}(1-x^{50}{)}^{101}dx$
$={\left[{\left(1-x^{50}\right)}^{101}\cdot x\right]}_0^1+\underset{0}{\overset{1}{\int }}{\left(1-x^{50}\right)}^{100}50\cdot x^{49}\cdot xdx$
[using integration by parts]
$=0-\underset{0}{\overset{1}{\int }}(50)(101)(1-x^{50}{)}^{100}({-x}^{50})dx$
$=-(50)(101)\underset{0}{\overset{1}{\int }}(1-x^{50}{)}^{101}dx$
$+(50)(101)\underset{0}{\overset{1}{\int }}(1-x^{50}{)}^{100}dx=5050I_2+5050I_1$
$\therefore I_2+5050I_2=5050I_1\Rightarrow \frac{(5050)I_2}{I_2}=5051$