A) \[\frac{1}{2550}\]
B) \[\frac{1}{2500}\]
C) \[\frac{10}{490}\]
D) \[\frac{1}{49}\]
Correct Answer: A
Solution :
Let \[I=\int_{0}^{1}{x{{(1-x)}^{49}}dx}\] \[=\int_{0}^{1}{(1-x)\,{{[1-(1-x)]}^{49}}\,dx}\] \[\left[ \because \,\,\int_{0}^{a}{f(x)\,dx=\int_{0}^{a}{f(a-x)\,dx}} \right]\] \[=\int_{0}^{1}{(1-x){{x}^{49}}\,dx}\] \[=\int_{0}^{1}{({{x}^{49}}-{{x}^{50}})dx}\] \[=\left[ \frac{{{x}^{50}}}{50}-\frac{{{x}^{51}}}{51} \right]_{0}^{1}\] \[=\left( \frac{1}{50}-\frac{1}{51} \right)-0=\frac{1}{50\times 51}=\frac{1}{2550}\]
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