Question

Calcule a integral por substituição de variável

Answer 1

Olá lucas:

Façamos,

$u = 2x$

Derivando implicitamente em u:

$\\ \frac{d}{du} (u) = \frac{d}{du} (2x) \\ \\ 1 = 2 \frac{dx}{du} \\ \\ 1du = 2dx \\ \\ \frac{du}{2} = dx$
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Substituindo-se

$\\ dx = \frac{du}{2} \\ u = 2x$
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$\\ \int\limits Cos2x {} \, dx = \int\limits Cosu {} \, \frac{du}{2} \\ \\ = \frac{1}{2} \int\limits Cosu {} \, du$

Lembrando que:

$\frac{d}{dx} ( Senx) = Cosx$

Então,

$\\ \frac{1}{2} \int\limits Cosu {} \, du = \frac{1}{2} Senu$

Agora, basta substituir "u = 2x" e colocarmos a constante "K" no final, Assim:


$\\ \frac{1}{2} Sen2x + K$

Answer 2

$\sf \displaystyle \int cos \left(2x\right)dx\\\\\\=\int cos \left(u\right)\frac{1}{2}du\\\\\\=\frac{1}{2}\cdot \int cos \left(u\right)du\\\\\\=\frac{1}{2}sin \left(u\right)\\\\\\=\frac{1}{2}sin \left(2x\right)\\\\\\\to \boxed{\sf =\frac{1}{2}sin \left(2x\right)+C}$