Question

The value of the integral $ \int_{0}^{1}{x{{(1-x)}^{49}}\,dx} $ is equal to

• A. $ \frac{1}{2550} $
• B. $ \frac{1}{2500} $
• C. $ \frac{10}{490} $
• D. $ \frac{1}{49} $

Answer 1

Answer: (A)

Let $ I=\int_{0}^{1}{x{{(1-x)}^{49}}dx} $
$ =\int_{0}^{1}{(1-x)\,{{[1-(1-x)]}^{49}}\,dx} $
$ \left[ \because \,\,\int_{0}^{a}{f(x)\,dx=\int_{0}^{a}{f(a-x)\,dx}} \right] $
$ =\int_{0}^{1}{(1-x){{x}^{49}}\,dx} $
$ =\int_{0}^{1}{({{x}^{49}}-{{x}^{50}})dx} $
$ =\left[ \frac{{{x}^{50}}}{50}-\frac{{{x}^{51}}}{51} \right]_{0}^{1} $
$ =\left( \frac{1}{50}-\frac{1}{51} \right)-0=\frac{1}{50\times 51}=\frac{1}{2550} $