Question

If $A={\int }_0^1{x}^{50}(2-x{)}^{50}dx;B={\int }_0^1{x}^{50}(1-x{)}^{50}dx$, which of the following is true?

• A. $A=2^{-100}B$
• B. $A=2^{50}B$
• C. $A=2^{100}B$
• D. None

Answer 1

The correct option is C $A=2^{100}B$

$A={\int }_0^1{x}^{50}(2-x{)}^{50}dx;B={\int }_0^1{x}^{50}(1-x{)}^{50}dx$

Let $x=2t$

$dx=2dt$

$\therefore A={\int }_0^{1/2}{2}^{101}{x}^{50}(1-x{)}^{50}dx.....(10$

$B=2{\int }_0^{1/2}{x}^{50}(1-x{)}^{50}dx........(2)$

From $\left(1\right)$ and $\left(2\right)$,

$\frac{A}{2^{101}}={\int }_0^{1/2}{x}^{50}(1-x{)}^{50}dx........(3)$

$\frac{B}{2}={\int }_0^{1/2}{x}^{50}(1-x{)}^{50}dx........(4)$

$\therefore \left(3\right)=\left(4\right)\Rightarrow \frac{A}{{20}^{101}}=\frac{B}{2}$

$\Rightarrow A=2^{100}B$