A) \[\int\limits_{0}^{1}{{{x}^{n-1}}{{(1-x)}^{n}}dx}\]
B) \[\int\limits_{0}^{1}{{{x}^{n}}{{(x-1)}^{n-1}}dx}\]
C) \[\int\limits_{0}^{1}{{{x}^{n-1}}{{(1+x)}^{n}}dx}\]
D) \[\int\limits_{0}^{1}{{{(1-x)}^{n}}{{x}^{n-1}}dx}\]
Correct Answer: B
Solution :
[b] Let, \[S=\frac{^{n}{{C}_{0}}}{n}+\frac{^{n}{{C}_{1}}}{n+1}+\frac{^{n}{{C}_{2}}}{n+2}+...+\frac{^{n}{{C}_{n}}}{2n}\] \[={{\left( \frac{1}{x} \right)}^{r}}{{=}^{m}}{{C}_{r}}{{x}^{2m-3r}}{{\,}^{n}}{{C}_{1}}\int\limits_{0}^{1}{{{x}^{n}}dx+...{{+}^{n}}{{C}_{n}}\int\limits_{0}^{2}{{{x}^{2n-1}}dx}}\] \[=\int\limits_{0}^{1}{{{[}^{n}}{{C}_{0}}{{x}^{n-1}}{{+}^{n}}{{C}_{1}}{{x}^{n}}+...{{+}^{n}}{{C}_{n}}{{x}^{2n-1}}]dx}\] \[={{\left( \frac{1}{x} \right)}^{r}}{{=}^{m}}{{C}_{r}}{{x}^{2m-3r}}\] \[=\int\limits_{1}^{2}{{{x}^{n}}{{(x-1)}^{n-1}}dx}\]
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