tanx dx
sec dx
The antiderivative (indefinite integral) of tan(x) can be found using substitution. Let u = cos(x), then du/dx = -sin(x) and dx = du/-sin(x). Substituting these values in the integral:
∫tan(x) dx = ∫(sin(x)/cos(x)) dx
= ∫(-du/u) (substituting sin(x) and dx)
= -ln|u| + C (using the antiderivative of u)
Now substituting u = cos(x) gives us:
∫tan(x) dx = -ln|cos(x)| + C
Therefore, the antiderivative of tan(x) is -ln|cos(x)| + C, where C is the constant of integration.
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