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 useful for readers.

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