Step by step solution to "Evaluate the definite integral exactly ∫01(e5x−5x)3(e5x−1)dx"

kramtus51

Answered question

2022-01-04

Evaluate the definite integral exactly

\int_{0}^{1} {(e^{5 x} - 5 x)}^{3} (e^{5 x} - 1) d x

Steve Hirano

Beginner2022-01-05Added 34 answers

Given

\int_{0}^{1} {(e^{5 x} - 5 x)}^{3} (e^{5 x} - 1) d x

[\begin{matrix} Let t = e^{5 x} - 5 x \\ \Rightarrow \frac{d t}{d x} = e^{5 x} (5) - 5 \\ \Rightarrow \frac{d t}{d x} = 5 (e^{5 x} - 1) \\ \Rightarrow \frac{d t}{5} = e^{5 x - 1} d x \end{matrix}]

and

[\begin{matrix} x = 0 \\ \Rightarrow t = e^{5 (0)} - 5 (0) \\ \Rightarrow t = 1 \\ x = 1 \Rightarrow t = e^{5 (1)} - 5 (1) \\ t = e^{5} - 5 \end{matrix}]

∴ \int_{0}^{1} {(e^{5 x} - 5 x)}^{3} (e^{5 x} - 1) d x = \int_{t = 1}^{e^{5} - 5} t^{3} \frac{d t}{5}

= \frac{1}{5} {[\frac{t^{4}}{4}]}_{t = 1}^{e^{5} - 5}

= \frac{1}{5} [\frac{{(e^{5} - 5)}^{4}}{4} - \frac{{(1)}^{4}}{4}]

= \frac{1}{20} [{(e^{5} - 5)}^{4} - 1]

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