Question and Answers Forum

All Questions      Topic List

Algebra Questions

Previous in All Question      Next in All Question      

Previous in Algebra      Next in Algebra      

Question Number 144359 by mathdanisur last updated on 24/Jun/21

∫ ((x^4 e^x  dx)/((x^4 +4x^3 +12x^2 +24x+24+72e^x )^2 )) = ?

$$\int\:\frac{{x}^{\mathrm{4}} {e}^{{x}} \:{dx}}{\left({x}^{\mathrm{4}} +\mathrm{4}{x}^{\mathrm{3}} +\mathrm{12}{x}^{\mathrm{2}} +\mathrm{24}{x}+\mathrm{24}+\mathrm{72}{e}^{{x}} \right)^{\mathrm{2}} }\:=\:? \\ $$

Answered by mitica last updated on 24/Jun/21

f(x)=x^4 +4x^3 +12x^2 +24x+24+72e^x   f′(x)=4x^3 +12x^2 +24x+24+72e^x   f(x)−f′(x)=x^4   ∫((f(x)−f′(x))/(f^2 (x)))e^x dx=∫(e^x /(f(x)))dx+∫((1/(f(x))))′e^x =  ∫(e^x /(f(x)))+e^x ∙(1/(f(x)))−∫(e^x /(f(x)))dx=(e^x /(f(x)))+c

$${f}\left({x}\right)={x}^{\mathrm{4}} +\mathrm{4}{x}^{\mathrm{3}} +\mathrm{12}{x}^{\mathrm{2}} +\mathrm{24}{x}+\mathrm{24}+\mathrm{72}{e}^{{x}} \\ $$$${f}'\left({x}\right)=\mathrm{4}{x}^{\mathrm{3}} +\mathrm{12}{x}^{\mathrm{2}} +\mathrm{24}{x}+\mathrm{24}+\mathrm{72}{e}^{{x}} \\ $$$${f}\left({x}\right)−{f}'\left({x}\right)={x}^{\mathrm{4}} \\ $$$$\int\frac{{f}\left({x}\right)−{f}'\left({x}\right)}{{f}^{\mathrm{2}} \left({x}\right)}{e}^{{x}} {dx}=\int\frac{{e}^{{x}} }{{f}\left({x}\right)}{dx}+\int\left(\frac{\mathrm{1}}{{f}\left({x}\right)}\right)'{e}^{{x}} = \\ $$$$\int\frac{{e}^{{x}} }{{f}\left({x}\right)}+{e}^{{x}} \centerdot\frac{\mathrm{1}}{{f}\left({x}\right)}−\int\frac{{e}^{{x}} }{{f}\left({x}\right)}{dx}=\frac{{e}^{{x}} }{{f}\left({x}\right)}+{c} \\ $$

Commented by mathdanisur last updated on 24/Jun/21

Thanks Sir, answer: x^4 .?

$${Thanks}\:{Sir},\:{answer}:\:{x}^{\mathrm{4}} .? \\ $$

Commented by mathdanisur last updated on 24/Jun/21

alot perfect solution thank you Sir

$${alot}\:{perfect}\:{solution}\:{thank}\:{you}\:{Sir} \\ $$

Commented by mitica last updated on 24/Jun/21

(e^x /(f(x)))

$$\frac{{e}^{{x}} }{{f}\left({x}\right)} \\ $$

Terms of Service

Privacy Policy

Contact: info@tinkutara.com