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Question Number 154823 by talminator2856791 last updated on 21/Sep/21

                        ∫_0 ^( ∞)  (e^(−x) /( x^(3/4)  )) dx

$$\: \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\int_{\mathrm{0}} ^{\:\infty} \:\frac{{e}^{−{x}} }{\:{x}^{\frac{\mathrm{3}}{\mathrm{4}}} \:}\:{dx} \\ $$$$\: \\ $$

Answered by Jonathanwaweh last updated on 21/Sep/21

                     ∫_0 ^( ∞)  (e^(−x) /( x^(3/4)  )) dx=∫_0 ^∞ x^(−(3/4)) e^(−x) dx=∫_0 ^∞ x^(1/4−1) e^(−(3/4)) dx=Γ(1/4)

$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\int_{\mathrm{0}} ^{\:\infty} \:\frac{{e}^{−{x}} }{\:{x}^{\frac{\mathrm{3}}{\mathrm{4}}} \:}\:{dx}=\int_{\mathrm{0}} ^{\infty} {x}^{−\frac{\mathrm{3}}{\mathrm{4}}} {e}^{−{x}} {dx}=\int_{\mathrm{0}} ^{\infty} {x}^{\mathrm{1}/\mathrm{4}−\mathrm{1}} {e}^{−\frac{\mathrm{3}}{\mathrm{4}}} {dx}=\Gamma\left(\mathrm{1}/\mathrm{4}\right) \\ $$$$ \\ $$

Commented by puissant last updated on 21/Sep/21

Mr jonathan, c′est :   ∫_0 ^∞ x^((1/4)−1) e^(−x) dx=Γ((1/4))=((π(√2))/(Γ((3/4))))..

$${Mr}\:{jonathan},\:{c}'{est}\:: \\ $$$$\:\int_{\mathrm{0}} ^{\infty} {x}^{\frac{\mathrm{1}}{\mathrm{4}}−\mathrm{1}} {e}^{−{x}} {dx}=\Gamma\left(\frac{\mathrm{1}}{\mathrm{4}}\right)=\frac{\pi\sqrt{\mathrm{2}}}{\Gamma\left(\frac{\mathrm{3}}{\mathrm{4}}\right)}.. \\ $$

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