Question

${\int }_0^{3}x{(3-x)}^{999}dx$ equals

• A. $\frac{3^{1001}}{\left(1000\right)\left(1001\right)}$
• B. $\frac{3^{1000}}{999\times 1000}$
• C. $\frac{3^{1000}}{\left(1000\right)\times \left(1001\right)}$
• D. None of these

Answer 1

Answer: (A)

$\frac{3^{1001}}{\left(1000\right)\left(1001\right)}$

We know that, $\int x^{n}dx=\frac{x^{n+1}}{n+1}+c$

${\int }_0^{3}x{(3-x)}^{999}dx$
$={[-\frac{x{(3-x)}^{1000}}{1000}]}_0^3+{\int }_0^3\frac{{(3-x)}^{1000}}{1000}dx$ ........ integration by parts
$={[0-\frac{{(3-x)}^{1001}}{1000\times 1001}]}_0^3=-0+\frac{3^{1001}}{1000\times 1001}=\frac{3^{1001}}{1000\times 1001}$