We can solve the integral of tan(x) with respect to x as follows:
First, we use the identity:
tan(x) = sin(x) / cos(x)
Then, we substitute u = cos(x), so du/dx = -sin(x) and dx = du / (-sin(x)).
The integral becomes:
∫ tan(x) dx = ∫ sin(x) / cos(x) dx
Substituting u and dx:
= ∫ -(1/u) du
= -ln|u| + C
= -ln|cos(x)| + C
Therefore, the final answer to the integral is:
∫ tan(x) dx = -ln|cos(x)| + C, where C is the constant of integration.
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