What is the integral of tan(1/2 x)?

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

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