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. $A=2^{50}B$
• B. $A=2^{-50}B$
• C. $A=2^{100}B$
• D. $A=2^{-100}B$

Answer 1

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

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

$\mathrm{In}A,\mathrm{put}x=2t\Rightarrow dx=2dt$

$\Rightarrow A=2{\int }_0^{1/2}{2}^{50}\cdot t^{50}{2}^{50}(1-t{)}^{50}dt........(i)$

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

$\left[\mathrm{using}{\int }_0^{2a}f\right(x)dx=2{\int }_0^{a}f(x\left)dx\text{if}f\right(2a-x)=f(x\left)\right]$

From equations $\left(1\right)$ and $\left(2\right),$ we get

$A=2^{100}B$