Question and Answers Forum

All Questions      Topic List

Differentiation Questions

Previous in All Question      Next in All Question      

Previous in Differentiation      Next in Differentiation      

Question Number 1571 by 112358 last updated on 20/Aug/15

∫_0 ^( ∞) (e^(−3x^2 ) /x^(1/5) )dx=?

$$\int_{\mathrm{0}} ^{\:\infty} \frac{{e}^{−\mathrm{3}{x}^{\mathrm{2}} } }{{x}^{\mathrm{1}/\mathrm{5}} }{dx}=? \\ $$

Answered by prakash jain last updated on 20/Aug/15

3x^2 =u x=((u/3))^(1/2) ⇒x^(1/5) =((u/3))^(1/10) 6xdx=du⇒dx=(du/(6x))=(du/(u^(1/2) 3^(1/2) 2)) (dx/x^(1/5) )= (du/(2(√(3u)))) ∙ (3^(1/10) /u^(1/10) )= (du/(2∙3^(2/5) ))∙u^(−3/5) =(du/3^(2/5) )u^((2/5)−1) ∫_0 ^( ∞) x^(t−1) e^(−x) dx=Γ(t) ∫_0 ^∞ u^((2/5) −1) e^(−u) du=Γ((2/5))

$$\mathrm{3}{x}^{\mathrm{2}} ={u} \\ $$$${x}=\left(\frac{{u}}{\mathrm{3}}\right)^{\mathrm{1}/\mathrm{2}} \Rightarrow{x}^{\mathrm{1}/\mathrm{5}} =\left(\frac{{u}}{\mathrm{3}}\right)^{\mathrm{1}/\mathrm{10}} \\ $$$$\mathrm{6}{xdx}={du}\Rightarrow{dx}=\frac{{du}}{\mathrm{6}{x}}=\frac{{du}}{{u}^{\mathrm{1}/\mathrm{2}} \mathrm{3}^{\mathrm{1}/\mathrm{2}} \mathrm{2}} \\ $$$$\frac{{dx}}{{x}^{\mathrm{1}/\mathrm{5}} }=\:\frac{{du}}{\mathrm{2}\sqrt{\mathrm{3}{u}}}\:\centerdot\:\frac{\mathrm{3}^{\mathrm{1}/\mathrm{10}} }{{u}^{\mathrm{1}/\mathrm{10}} }=\:\:\frac{{du}}{\mathrm{2}\centerdot\mathrm{3}^{\mathrm{2}/\mathrm{5}} }\centerdot{u}^{−\mathrm{3}/\mathrm{5}} \:=\frac{{du}}{\mathrm{3}^{\mathrm{2}/\mathrm{5}} }{u}^{\frac{\mathrm{2}}{\mathrm{5}}−\mathrm{1}} \\ $$$$\int_{\mathrm{0}} ^{\:\infty} {x}^{{t}−\mathrm{1}} {e}^{−{x}} {dx}=\Gamma\left({t}\right) \\ $$$$\int_{\mathrm{0}} ^{\infty} \:{u}^{\frac{\mathrm{2}}{\mathrm{5}}\:−\mathrm{1}} {e}^{−{u}} {du}=\Gamma\left(\frac{\mathrm{2}}{\mathrm{5}}\right) \\ $$

Terms of Service

Privacy Policy

Contact: info@tinkutara.com