Question

If $I={\int }_0^{1}x(1-x{)}^{2007/2}dx,$ then $I$ equals

• A. $2007/2$
• B. $2011/2$
• C. $2009/2$
• D. none of these

Answer 1

The correct option is D none of these

Using ${\int }_0^{a}f\left(x\right)dx={\int }_0^{a}f(a-x)dx,$ we get
$I={\int }_0^1(1-x)x^{2007/2}dx$
$={(\frac{2}{2009}{x}^{2009/2}-\frac{2}{2011}{x}^{2011/2})]}_0^1$
$=\frac{4}{\left(2009\right)\left(2011\right)}$

TIP For $0\le x\le 1,$
$0\le x(1-x{)}^{2007/2}\le 1,$
$\Rightarrow 0\le {\int }_0^{1}x(1-x{)}^{2007/2}dx\le {\int }_0^{1}dx=1$.

Therefore, answers (a), (b) and (c) can be ruled out,