Question

∫e^2x senx x resolução​

Answer 1

Se for ∫e^(2x)  senx  dx

Fazendo por partes

u =sen(x)   ==> du =cos(x) dx

e^(2x) dx =dv  ==>  ∫e^(2x) dx = ∫ dv  ==>(1/2)*e^(2x) = v

∫e^(2x)  senx  dx = sen(x) * (1/2)*e^(2x)  -   ∫  (1/2)*e^(2x) * cos(x) dx

∫e^(2x)  senx  dx = (1/2)* sen(x) * e^(2x)  -(1/2)  ∫  e^(2x) * cos(x) dx

___________________________________________________

*****∫  e^(2x) * cos(x) dx

Fazendo por partes

u =cos(x) ==>du =-sen(x) dx

e^(2x) dx =dv  ==>  ∫e^(2x) dx = ∫ dv  ==>(1/2)*e^(2x) = v

∫  e^(2x) * cos(x) dx  = cos(x)* (1/2)*e^(2x)  - ∫ (1/2)*e^(2x) (-sen(x)) dx

∫  e^(2x) * cos(x) dx  = (1/2)*cos(x)*e^(2x)  + (1/2)∫ e^(2x) *sen(x) dx

___________________________________________________

∫e^(2x)  senx  dx = (1/2)* sen(x) * e^(2x)  -(1/2) [(1/2)*cos(x)*e^(2x)  + (1/2)∫ e^(2x) *sen(x) dx]

∫e^(2x)  senx  dx = (1/2)* sen(x) * e^(2x)  - (1/4)*cos(x)*e^(2x)  -(1/4)∫ e^(2x) *sen(x) dx

∫e^(2x)  senx  dx +(1/4)∫ e^(2x) *sen(x) dx = (1/2)* sen(x) * e^(2x)  - (1/4)*cos(x)*e^(2x)

∫e^(2x)  senx  dx * [1+1/4] = (1/2)* sen(x) * e^(2x)  - (1/4)*cos(x)*e^(2x)

∫e^(2x)  senx  dx * [5/4] = (1/2)* sen(x) * e^(2x)  - (1/4)*cos(x)*e^(2x)

∫e^(2x)  senx  dx  = (2/5)* sen(x) * e^(2x)  - (1/5)*cos(x)*e^(2x)

∫e^(2x)  senx  dx  =(1/5)* e^(2x) * [2sen(x)-cos(x)]  + constante