Question

$\int\limits 2senx/cosx \, dx$

Answer 1

$\int\limits { \frac{2*sen(x)}{cos(x)} } \, dx = \boxed{\boxed{2 \int { \frac{sen(x)}{cos(x)} } \, dx }}$

fazendo a substituiçao
$u = cos(x)$

derivando u em relaçao a x
$\frac{du}{dx}= -sen(x) \\\\ \frac{du}{-sen(x)} =dx$

substituindo na integral
$2 \int { \frac{sen(x)}{u } * \frac{du}{-sen(x)} }\\\\ = 2 \int\limits { \frac{-1}{u} } \, du \\\\ = 2*-ln(u)\\\\=-2ln(u)$

como u = cos (x)

$\boxed{\boxed{ \int\limits { \frac{2*sen(x)}{cos(x)} } \, dx=-2*ln(cos(x))+C}}$

Answer 2

$\large\mathsf{\int(2\dfrac{\sin(x)}{\cos(x)})dx}\\\mathsf{2\int\tan(x)dx=2\int\dfrac{\sec(x)</p><p>.\tan(x)}{\sec(x)}dx} \\=\huge\mathsf{2 ln| \sec(x) | } + c$