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

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= - \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