Evaluate \(\rm \int_{0}^{1}{e}^{{x}}({e}^{{x}}-1)^{4}dx\)

  1. \(\rm \frac{(e+1)^{5}}{5}\)
  2. \(\rm \frac{(e-1)^{5}}{4}\)
  3. \(\rm \frac{(e-1)^{5}}{5}\)
  4. 0

Answer (Detailed Solution Below)

Option 3 : \(\rm \frac{(e-1)^{5}}{5}\)
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Detailed Solution

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Concept:

\(\rm \int x^n dx=\frac{x^{n+1}}{n+1}+c\)

 

Calculation:

Let I = \(\rm \int_{0}^{1}{e}^{{x}}({e}^{{x}}-1)^{4}dx\)

Let (ex - 1) = t

Differentiating with respect to x, we get

⇒ exdx = dt

\(\rm I = \int_{0}^{1} t^{4}dt \\=\left[\frac{t^{5}}{5}\right]_0^1\)

Put t = (ex - 1) 

\(\rm =\left[\frac{(e^x-1)^{5}}{5}\right]_0^1\\=\left[\frac{(e^1-1)^{5}}{5} -\frac{(e^0-1)^{5}}{5} \right]\\=\frac{(e-1)^{5}}{5}\)

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