Mathematics

Evaluate: $$\displaystyle \int _{ 0 }^{ 1 }{ { x\left( 1-x \right)  }^{ n } } dx$$


SOLUTION
$$I=\displaystyle \int_{0}^{1}x(1-x)^{n}dx$$
$$=\displaystyle \int_{0}^{1}(1-x)(1-(1-x))^{n}dx$$ [ Using property of definite integral]
$$=\displaystyle \int_{0}^{1}(1-x)(x)^{n}dx$$
$$=\displaystyle \int_{0}^{1}x^{n}-x^{x-1}dx$$
$$=\displaystyle \left [ \dfrac{x^{n+1}}{n+1} -\dfrac{x^{n+2}}{n+2}\right ]_{0}^{1}$$
$$=\dfrac{1}{n+1}-\dfrac{1}{n+2}=\dfrac{1}{(n+1)(n+2)}$$
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Subjective Medium Published on 17th 09, 2020
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