Question

The value of $\frac{\left(5050\right){\int }_0^1{(1-x^{50})}^{100}dx}{{\int }_0^1{(1-x^{50})}^{101}dx}$ is

• A. $5051$
• B. $5052$
• C. $2525$
• D. $2526$

Answer 1

The correct option is A $5051$

Let $I_2={\int }_0^1{(1-x^{50})}^{101}dx$

Using integration by parts

$={[{(1-x^{50})}^{101}.x]}_0^1+{\int }_0^1{(1-x^{50})}^{100}50.x^{49}xdx$

$=0-{\int }_0^1\left(50\right)\left(101\right){(1-x^{50})}^{100}\left({-x}^{50}\right)dx$

$=-\left(50\right)\left(101\right){\int }_0^1{\left({1-x}^{50}\right)}^{101}dx+\left(50\right)\left(101\right){\int }_0^1{\left({1-x}^{50}\right)}^{100}dx$

$={5050I}_2+{5050I}_1$

$\therefore I_2+{5050I}_2={5050I}_1$

$\therefore \frac{\left(5050\right)I_1}{I_2}=5051$