What is the integral of 1/(sin(0.5x)cos(0.5x)) ?

The integral of 1/(sin(0.5x)cos(0.5x)) is 2ln|tan(0.5x)|+C

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integral of 1/(sin(0.5x)cos(0.5x))

\int \frac{1}{sin ( 0.5 x ) cos ( 0.5 x )} d x

2 ln ∣ tan (0.5 x) ∣ + C

Solution steps

\int \frac{1}{sin ( 0.5 x ) cos ( 0.5 x )} d x

Apply u-substitution

: \int \frac{1}{0.5 sin ( u ) cos ( u )} d u

= \int \frac{1}{0.5 sin ( u ) cos ( u )} d u

Take the constant out:

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

= \frac{1}{0.5} \cdot \int \frac{1}{sin ( u ) cos ( u )} d u

Simplify

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

Rewrite using trig identities

= 2 \cdot \int \frac{sec ( u )}{sin ( u )} d u

Apply Trigonometric Substitution

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

= 2 \cdot \int \frac{1}{v} d v

Use the common integral:

\int \frac{1}{v} d v = ln (∣ v ∣)

= 2 ln ∣ v ∣

Substitute back

= 2 ln ∣ tan (0.5 x) ∣

Add a constant to the solution

= 2 ln ∣ tan (0.5 x) ∣ + C

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What is the integral of 1/(sin(0.5x)cos(0.5x)) ?
The integral of 1/(sin(0.5x)cos(0.5x)) is 2ln|tan(0.5x)|+C