Question and Answers Forum

All Questions      Topic List

Others Questions

Previous in All Question      Next in All Question      

Previous in Others      Next in Others      

Question Number 91302 by 174 last updated on 29/Apr/20

Answered by MJS last updated on 29/Apr/20

∫e^x (((x−1)^2 )/((x^2 +1)^2 ))dx= =∫(e^x /(x^2 +1))dx−2∫((e^x x)/((x^2 +1)^2 ))dx the 2^(nd) one by parts: −2∫((e^x x)/((x^2 +1)^2 ))dx= u′=(x/((x^2 +1)^2 )) → u=−(1/(2(x^2 +1))) v=v′=e^x =(e^x /(x^2 +1))−∫(e^x /(x^2 +1))dx so we have ∫(e^x /(x^2 +1))dx+(e^x /(x^2 +1))−∫(e^x /(x^2 +1))dx= =(e^x /(x^2 +1))+C

$$\int\mathrm{e}^{{x}} \frac{\left({x}−\mathrm{1}\right)^{\mathrm{2}} }{\left({x}^{\mathrm{2}} +\mathrm{1}\right)^{\mathrm{2}} }{dx}= \\ $$$$=\int\frac{\mathrm{e}^{{x}} }{{x}^{\mathrm{2}} +\mathrm{1}}{dx}−\mathrm{2}\int\frac{\mathrm{e}^{{x}} {x}}{\left({x}^{\mathrm{2}} +\mathrm{1}\right)^{\mathrm{2}} }{dx} \\ $$$$\mathrm{the}\:\mathrm{2}^{\mathrm{nd}} \:\mathrm{one}\:\mathrm{by}\:\mathrm{parts}: \\ $$$$−\mathrm{2}\int\frac{\mathrm{e}^{{x}} {x}}{\left({x}^{\mathrm{2}} +\mathrm{1}\right)^{\mathrm{2}} }{dx}= \\ $$$$\:\:\:\:\:{u}'=\frac{{x}}{\left({x}^{\mathrm{2}} +\mathrm{1}\right)^{\mathrm{2}} }\:\rightarrow\:{u}=−\frac{\mathrm{1}}{\mathrm{2}\left({x}^{\mathrm{2}} +\mathrm{1}\right)} \\ $$$$\:\:\:\:\:{v}={v}'=\mathrm{e}^{{x}} \\ $$$$=\frac{\mathrm{e}^{{x}} }{{x}^{\mathrm{2}} +\mathrm{1}}−\int\frac{\mathrm{e}^{{x}} }{{x}^{\mathrm{2}} +\mathrm{1}}{dx} \\ $$$$\mathrm{so}\:\mathrm{we}\:\mathrm{have} \\ $$$$\int\frac{\mathrm{e}^{{x}} }{{x}^{\mathrm{2}} +\mathrm{1}}{dx}+\frac{\mathrm{e}^{{x}} }{{x}^{\mathrm{2}} +\mathrm{1}}−\int\frac{\mathrm{e}^{{x}} }{{x}^{\mathrm{2}} +\mathrm{1}}{dx}= \\ $$$$=\frac{\mathrm{e}^{{x}} }{{x}^{\mathrm{2}} +\mathrm{1}}+\mathrm{C} \\ $$

Answered by $@ty@m123 last updated on 29/Apr/20

Formula: ∫e^x {f(x)+f ′(x)}dx=e^x f(x)+C Here, f(x)=(1/(1+x^2 ))

$${Formula}: \\ $$$$\int{e}^{{x}} \left\{{f}\left({x}\right)+{f}\:'\left({x}\right)\right\}{dx}={e}^{{x}} {f}\left({x}\right)+\mathrm{C} \\ $$$${Here}, \\ $$$${f}\left({x}\right)=\frac{\mathrm{1}}{\mathrm{1}+{x}^{\mathrm{2}} } \\ $$

Terms of Service

Privacy Policy

Contact: info@tinkutara.com