Question

Qual a integral de cos3x- sen3x ?

Answer 1

$I(x) = \int\ [cos(3x) - sen(3x)] \, dx = \int\ cos(3x)\,dx - \int\ sen(3x)\, dx \\ \\ \int\ cos(3x)\,dx \Rightarrow t = 3x \Longleftrightarrow dt = 3dx \Longleftrightarrow dx = \dfrac{1} {3}dt \\ \\ \int\ cos(t)\, \dfrac{1}{3} dt = \dfrac{1}{3}\int\ cos(t)\, dt = \dfrac{1}{3} sen(t) + C1 \\ \\ \therefore \int\ cos(3x)\,dx = \dfrac{1}{3} sen(3x) + C1 \ \ \ \ \ \ (I)$
$\int\ sen(3x)\, dx \Rightarrow t = 3x \Longleftrightarrow dt = 3dx \Longleftrightarrow dx = \dfrac{1}{3}dt \\ \\ \int\ sen(t)\, \dfrac{1}{3} dt = -\dfrac{1}{3} cos(t) + C2 \\ \\ \therefore \int\ sen(3x)\, dx =-\dfrac{1}{3} cos(3x) + C2 \ \ \ \ \ \ \ \ (II) \\ \\ Somando \ (I) \ com \ (II) \ fica: \\ \\ I(x) = \dfrac{1}{3} [sen(3x) + cos(3x)] + C \\ \\ \Longrightarrow C1 + C2 = C \ \ a \ constante \ de \ \text{integra{\c c}{\~ a}o}$

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Obrigado pela oportunidade. 
Boa sorte, bons estudos.
SSRC - 2015 
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