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

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

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integral of 1/(sin(x/2)cos(x/2))

\int \frac{1}{sin ( \frac{x}{2} ) cos ( \frac{x}{2} )} d x

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

Solution steps

\int \frac{1}{sin ( \frac{x}{2} ) cos ( \frac{x}{2} )} d x

Rewrite using trig identities

= \int \frac{2}{sin ( x )} d x

Take the constant out:

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

= 2 \cdot \int \frac{1}{sin ( x )} d x

Apply u-substitution

: \int \frac{1}{u} d u

= 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 = tan (\frac{x}{2})

= 2 ln tan (\frac{x}{2})

Add a constant to the solution

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

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