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

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

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integral of (2ln(tan(x)))/(sin(x)cos(x))

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

ln^{2} (tan (x)) + C

Solution steps

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

Apply Trigonometric Substitution

: \int \frac{ln ( u )}{u} d u

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

Apply u-substitution

: \int v d v

= 2 \cdot \int v d v

Apply the Power Rule

: \frac{v ^{2}}{2}

= 2 \cdot \frac{v ^{2}}{2}

Substitute back

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

Simplify

2 \cdot \frac{ln ^{2} ( tan ( x ) )}{2} : ln^{2} (tan (x))

= ln^{2} (tan (x))

Add a constant to the solution

= ln^{2} (tan (x)) + C

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