To find the integral of tan(1/2 x), we start by using a substitution method to simplify the expression. Let us set:
u = 1/2 x
This implies that:
du = (1/2) dx, or dx = 2 du
Now, we can rewrite the integral:
∫ tan(1/2 x) dx = ∫ tan(u) * 2 du
This simplifies to:
2 ∫ tan(u) du
The integral of tan(u) can be solved using the identity:
tan(u) = sin(u) / cos(u)
The integral of tan(u) is:
∫ tan(u) du = -ln|cos(u)| + C
Substituting this back, we have:
2 ∫ tan(u) du = -2 ln|cos(u)| + C
Now substituting back for u, we get:
-2 ln|cos(1/2 x)| + C
Thus, the final answer for the integral of tan(1/2 x) is:
∫ tan(1/2 x) dx = -2 ln|cos(1/2 x)| + C