Question Number 172012 by Mikenice last updated on 23/Jun/22
$${find} \\ $$$$\int{e}^{{x}} {sinxdx} \\ $$
Answered by puissant last updated on 23/Jun/22
$${J}=\int{e}^{{x}} {sinxdx} \\ $$$$\begin{cases}{{u}'={e}^{{x}} }\\{{v}={sinx}}\end{cases}\:\Rightarrow\:\begin{cases}{{u}={e}^{{x}} }\\{{v}'={cosx}}\end{cases} \\ $$$${J}\:=\:{e}^{{x}} {sinx}\:−\int{e}^{{x}} {cosxdx} \\ $$$${L}=\int{e}^{{x}} {cosxdx} \\ $$$$\begin{cases}{{u}'={e}^{{x}} }\\{{v}={cosx}}\end{cases}\:\Rightarrow\:\begin{cases}{{u}={e}^{{x}} }\\{{v}'=−{sinx}}\end{cases} \\ $$$${L}\:=\:{e}^{{x}} {cosx}\:+\:\int{e}^{{x}} {sinxdx} \\ $$$$\Rightarrow\:{J}=\:{e}^{{x}} {sinx}\:−\:{e}^{{x}} {cosx}−{J}\: \\ $$$$\Rightarrow\:{J}\:\:=\:\:\frac{{e}^{{x}} }{\mathrm{2}}{sinx}\:\:−\:\frac{{e}^{{x}} }{\mathrm{2}}{cosx}\:+\:{C} \\ $$