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Question Number 106809 by Study last updated on 07/Aug/20

∫e^(2x) sine^x dx=??

$$\int{e}^{\mathrm{2}{x}} {sine}^{{x}} {dx}=?? \\ $$

Answered by bemath last updated on 07/Aug/20

@bemath@ ∫ e^(x ) sin (e^x )d(e^x ) = ∫ w sin w dw = −wcos w −sin w + C =−e^x cos (e^x )−sin (e^x ) + C

$$\:\:\:\:\:@\mathrm{bemath}@ \\ $$$$\int\:\mathrm{e}^{\mathrm{x}\:} \mathrm{sin}\:\left(\mathrm{e}^{\mathrm{x}} \right)\mathrm{d}\left(\mathrm{e}^{\mathrm{x}} \right)\:=\:\int\:\mathrm{w}\:\mathrm{sin}\:\mathrm{w}\:\mathrm{dw} \\ $$$$=\:−\mathrm{wcos}\:\mathrm{w}\:−\mathrm{sin}\:\mathrm{w}\:+\:\mathrm{C} \\ $$$$=−\mathrm{e}^{\mathrm{x}} \:\mathrm{cos}\:\left(\mathrm{e}^{\mathrm{x}} \right)−\mathrm{sin}\:\left(\mathrm{e}^{\mathrm{x}} \right)\:+\:\mathrm{C}\: \\ $$

Commented by malwaan last updated on 07/Aug/20

−e^x cos(e^x )−sin(e^x )+ C thank you sir bemath

$$−{e}^{{x}} {cos}\left({e}^{{x}} \right)−{sin}\left({e}^{{x}} \right)+\:{C} \\ $$$${thank}\:{you}\:{sir}\:{bemath} \\ $$

Commented by bemath last updated on 07/Aug/20

oo yes. sorry typo

$$\mathrm{oo}\:\mathrm{yes}.\:\mathrm{sorry}\:\mathrm{typo} \\ $$

Answered by mathmax by abdo last updated on 07/Aug/20

I =∫ e^(2x) sin(e^x )dx we do tbe changement e^x =t ⇒ I =∫t^2 sin(t)(dt/t) =∫ tsint dt by parts u =t and v^′ =sint ⇒ I =−tcost +∫ cost dt =−tcost +sint +C =−e^x cos(e^x )+sin(e^x )+C

$$\mathrm{I}\:=\int\:\mathrm{e}^{\mathrm{2x}} \mathrm{sin}\left(\mathrm{e}^{\mathrm{x}} \right)\mathrm{dx}\:\mathrm{we}\:\mathrm{do}\:\mathrm{tbe}\:\mathrm{changement}\:\mathrm{e}^{\mathrm{x}} \:=\mathrm{t}\:\:\Rightarrow \\ $$$$\mathrm{I}\:=\int\mathrm{t}^{\mathrm{2}} \mathrm{sin}\left(\mathrm{t}\right)\frac{\mathrm{dt}}{\mathrm{t}}\:=\int\:\mathrm{tsint}\:\mathrm{dt}\:\:\mathrm{by}\:\mathrm{parts}\:\mathrm{u}\:=\mathrm{t}\:\mathrm{and}\:\mathrm{v}^{'} \:=\mathrm{sint}\:\Rightarrow \\ $$$$\mathrm{I}\:=−\mathrm{tcost}\:+\int\:\mathrm{cost}\:\mathrm{dt}\:=−\mathrm{tcost}\:+\mathrm{sint}\:+\mathrm{C} \\ $$$$=−\mathrm{e}^{\mathrm{x}} \:\mathrm{cos}\left(\mathrm{e}^{\mathrm{x}} \right)+\mathrm{sin}\left(\mathrm{e}^{\mathrm{x}} \right)+\mathrm{C} \\ $$

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