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

The integral of tan(1/2 x) is xtan(1/2 x)-xtan(x/2)-2ln|cos(x/2)|+C

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integral of tan(1/2 x)

\int tan (\frac{1}{2} x) d x

x tan (\frac{1}{2} x) - x tan (\frac{x}{2}) - 2 ln cos (\frac{x}{2}) + C

Solution steps

\int tan (\frac{1}{2} x) d x

Apply Integration By Parts

: tan (\frac{1}{2} x) x - \int sec^{2} (\frac{x}{2}) \frac{1}{2} x d x

= tan (\frac{1}{2} x) x - \int sec^{2} (\frac{x}{2}) \frac{1}{2} x d x

\int sec^{2} (\frac{x}{2}) \frac{1}{2} x d x = x tan (\frac{x}{2}) + 2 ln cos (\frac{x}{2})

= tan (\frac{1}{2} x) x - (x tan (\frac{x}{2}) + 2 ln cos (\frac{x}{2}))

Simplify

tan (\frac{1}{2} x) x - (x tan (\frac{x}{2}) + 2 ln cos (\frac{x}{2})) : x tan (\frac{1}{2} x) - x tan (\frac{x}{2}) - 2 ln cos (\frac{x}{2})

= x tan (\frac{1}{2} x) - x tan (\frac{x}{2}) - 2 ln cos (\frac{x}{2})

Add a constant to the solution

= x tan (\frac{1}{2} x) - x tan (\frac{x}{2}) - 2 ln cos (\frac{x}{2}) + C

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What is the integral of tan(1/2 x) ?
The integral of tan(1/2 x) is xtan(1/2 x)-xtan(x/2)-2ln|cos(x/2)|+C