Integration of tan x

Integration of tanx - In this paper, we will discuss how to integrate the following tan (x).

The technique used is the integration of substitution techniques.

$$ ∫ tan(x) \ dx = ∫ \frac{sin(x)}{cos(x)} \ dx $$

Notice the integran on the right, we can assume u = cos (x) because the derivatives of cos (x) are -sin (x).

If u = cos (x) then du = -sin (x) dx so we get:

$$ \begin{align} ∫ tan \ (x) \ dx &= ∫ \frac{sin \ (x)}{cos \ (x)} \ dx \\ &= -∫ \frac{du}{u} \\ &= -ln \ (u) \\ &= -ln \ (cos \ x) \end{align} $$

So, $ ∫ tan \ (x) \ dx = -ln \ (cos \ x) + C $

Note:

$ ∫ \frac{1}{x} \ dx = ln \ (x) + C $

ln: natural logarithm.

Hopefully this article, Integration of tanx, useful for readers.