Mathematics

Compute the integral$$\displaystyle \int_{0}^{1}\left ( e^{x}-1 \right )^{4}e^{x}dx$$


ANSWER

$$\displaystyle 0.2\left ( e-1 \right )^{5}$$


SOLUTION
Let $$I=\int _{ 0 }^{ 1 }{ { \left( { e }^{ x }-1 \right)  }^{ 4 } } { e }^{ x }dx\quad $$

Put $${ e }^{ x }-1=t\Rightarrow { e }^{ x }dx=dt$$

$$x=0\implies t=0$$ and $$x=1 \implies t=e-1$$

$$\displaystyle I=\int _{ 0 }^{ e-1 }{ { t }^{ 4 } } dt=\left[ \frac { { t }^{ 5 } }{ 5 }  \right] _{ 0 }^{ e-1 }{ = }\quad 0.2{ \left( e-1 \right)  }^{ 5 }$$
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Single Correct Medium Published on 17th 09, 2020
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