What is the integral of tan(t/2) ?

The integral of tan(t/2) is -2ln|cos(t/2)|+C

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integral of tan(t/2)

\int tan (\frac{t}{2}) d t

- 2 ln cos (\frac{t}{2}) + C

Solution steps

\int tan (\frac{t}{2}) d t

Rewrite using trig identities

= \int \frac{sin ( \frac{t}{2} )}{cos ( \frac{t}{2} )} d t

Apply u-substitution

: \int - \frac{2}{u} d u

= \int - \frac{2}{u} d u

Take the constant out:

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

= - 2 \cdot \int \frac{1}{u} d u

Use the common integral:

\int \frac{1}{u} d u = ln (∣ u ∣)

= - 2 ln ∣ u ∣

Substitute back

u = cos (\frac{t}{2})

= - 2 ln cos (\frac{t}{2})

Add a constant to the solution

= - 2 ln cos (\frac{t}{2}) + C

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