Mathematics

Integrate : $$\int\limits_0^1 {x{{\left( {1 - x} \right)}^{91}}} $$


SOLUTION
Let $$t=1-x\Rightarrow\,dt=-dx$$

When $$x=0\Rightarrow\,t=1$$
When $$x=1\Rightarrow\,t=0$$

$$=-\displaystyle\int_{1}^{0}{{t}^{91}\left(1-t\right)dt}$$

$$=-\displaystyle\int_{1}^{0}{{t}^{91}dt}+\displaystyle\int_{1}^{0}{{t}^{92}dt}$$

$$=\left[-\dfrac{{t}^{91+1}}{92}+\dfrac{{t}^{92+1}}{93}\right]_{1}^{0}$$

$$=\left[-\dfrac{{t}^{92}}{92}+\dfrac{{t}^{93}}{93}\right]_{1}^{0}$$

$$=-\dfrac{1}{92}\left(0-1\right)+\dfrac{1}{93}\left(0-1\right)$$

$$=\dfrac{1}{92}-\dfrac{1}{93}=\dfrac{93-92}{92\times 93}=\dfrac{1}{8,556‬}$$
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Subjective Medium Published on 17th 09, 2020
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