What is the integral of-4tan(x/2) ?

The integral of-4tan(x/2) is 8ln|cos(x/2)|+C

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integral of-4tan(x/2)

\int - 4 tan (\frac{x}{2}) d x

8 ln cos (\frac{x}{2}) + C

Solution steps

\int - 4 tan (\frac{x}{2}) d x

Take the constant out:

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

= - 4 \cdot \int tan (\frac{x}{2}) d x

Rewrite using trig identities

= - 4 \cdot \int \frac{sin ( \frac{x}{2} )}{cos ( \frac{x}{2} )} d x

Apply u-substitution

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

= - 4 \cdot \int - \frac{2}{u} d u

Take the constant out:

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

= - 4 (- 2 \cdot \int \frac{1}{u} d u)

Use the common integral:

\int \frac{1}{u} d u = ln (∣ u ∣)

= - 4 (- 2 ln ∣ u ∣)

Substitute back

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

= - 4 (- 2 ln cos (\frac{x}{2}))

Simplify

= 8 ln cos (\frac{x}{2})

Add a constant to the solution

= 8 ln cos (\frac{x}{2}) + C

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What is the integral of-4tan(x/2) ?
The integral of-4tan(x/2) is 8ln|cos(x/2)|+C