Question

integral de sen(x)/cos(x)

Answer 1

Calcular a integral indefinida:

    $\mathsf{\displaystyle\int \frac{sen\,x}{cos\,x}\,dx}\\\\\\ \mathsf{\displaystyle=\int \frac{1}{cos\,x}\cdot sen\,x\,dx}\\\\\\ \mathsf{\displaystyle=-\int \frac{1}{cos\,x}\cdot (-sen\,x)\,dx}$

   

Faça a seguinte mudança de variável:

    $\mathsf{u=cos\,x}\quad\Longrightarrow\quad \mathsf{du=-sen\,x\,dx}$

e a integral fica

    $\mathsf{\displaystyle=-\int \frac{1}{u}\,du}\\\\\\ \mathsf{\displaystyle=-\,\ell n\,|u|+C}\\\\\\ \mathsf{\displaystyle=-\,\ell n\,|cos\,x|+C\quad\longleftarrow\quad resposta.}$

Bons estudos! :-)

Answer 2

Resposta:

$\int\frac{sen(x)}{cos(x)}\,dx=-ln|cos(x)|+C$

Resolução:

$\int\frac{sen(x)}{cos(x)}\,dx$

Por substituição:

$u=cos(x)$

$du=-sen(x)dx$

Temos que:

$\int\frac{sen(x)}{cos(x)}\,dx$

$=-\int\frac{-sen(x)}{cos(x)}\,dx$

$=-\int\frac{du}{u}$

$=-ln|u|+C$

$=-ln|cos(x)|+C$