Q-0-x-c-c-x-dx-

Question Number 156470 by puissant last updated on 11/Oct/21

Q = \int_{0}^{\infty} \frac{x^{c}}{c^{x}} d x

Answered by puissant last updated on 11/Oct/21

p o s o n s c^{x} = e^{u} \Rightarrow e^{x l n c} = e^{u} \Rightarrow x = \frac{u}{l n c}

\Rightarrow d x = \frac{d u}{l n c}

\Rightarrow Q = \int_{0}^{\infty} \frac{{(\frac{u}{l n c})}^{c}}{e^{u}} \times \frac{d u}{l n c} = \frac{1}{{(l n c)}^{c + 1}} \int_{0}^{\infty} \frac{u^{c}}{e^{u}} d u

= \frac{1}{{(l n c)}^{c + 1}} \int_{0}^{\infty} u^{c - 1 + 1} e^{- u} d u

∴∵ Q = \int_{0}^{\infty} \frac{x^{c}}{c^{x}} d x = \frac{Γ (c + 1)}{{(l n c)}^{c + 1}} . .

\dots \dots \dots . L e p u i s s a n t \dots \dots \dots

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