Question

Calcule a integral: x^2 sen3x dx

Answer 1

Espero ajudar.

X³/3.-cos(x).3x²/2

Answer 2

∫x² Sen [3x] dx  

➊ Essa integral se resolve por partes:  

Formula:  

∫u dv = u v - ∫v du  

Onde:  

u = x² : : : : : : : : dv = Sen [3x]  

du = 2x dx : : : : v = - [1/3] Cos [3x]  

➋ Aplicamos a Formula:  

∫u dv = u v - ∫v du  

∫x² Sen [3x]dx = - [1/3] x² Cos [3x] - ∫ - [2/3] x Cos [3x] dx  

∫x² Sen [3x]dx = - [1/3] x² Cos [3x] + [2/3] ( ∫x Cos [3x] dx )  

➌ voltamos a aplicar a Formula:  

Onde:  

u = x : : : : : : dv = Cos [3x]  

du = dx : : : : v = [1/3] Sen [3x]  

∫u dv = u v - ∫v du  

∫x² Sen [3x]dx = - [1/3] x² Cos [3x] + [2/3]( [1/3]x sen [3x] - ∫ [1/3] Sen [3x] dx )  

➍ Integramos, desenvolvemos oque resta  

∫x² Sen [3x]dx = - [1/3] x² Cos [3x] + [2/3] ( [1/3] x sen [3x] + [1/3] [1/3] Cos [3x] )  

∫x² Sen [3x]dx = - [1/3] x² Cos [3x] + [2/9]x sen [3x] + [2/27] Cos [3x] + C  

Este é o resultado  

======================================...  

- [1/3] x² Cos [3x] + [2/9]x sen [3x] + [2/27] Cos [3x] + C  

======================================...