Mathematics - Definite Integration Question with Solution | TestHub

Let $I_2={\int }_0^1{\left(1-x^{50}\right)}^{101}dx$, using integration by parts $$\begin{array}{rl}& {={\left[(1-x{)}^{50}\right)}^{101}\cdot x]}_0^1+{\int }_0^1{\left(1-x^{50}\right)}^{100}50\cdot x^{49}\cdot xdx\\ & =0-{\int }_0^1(50)(101){\left(1-x^{50}\right)}^{100}(-x{)}^{50}dx\\ & =-(50)(101){\int }_0^1{\left(1-x^{50}\right)}^{101}dx+(50)(101){\int }_0^1{\left(1-x^{50}\right)}^{100}dx\\ & =5050I_2+5050I_1\end{array}$$ $$\begin{array}{rlr}\therefore & I_2+5050I_2& =5050I_1\\ \therefore & \frac{(5050)I_1}{I_2}& =5051\end{array}$$