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

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

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

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

\frac{1}{2} ln^{2} (tan (\frac{x}{2})) + C

Solution steps

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

Apply u-substitution

= \int 2 csc (2 u) ln (tan (u)) d u

Take the constant out:

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

= 2 \cdot \int csc (2 u) ln (tan (u)) d u

Rewrite using trig identities

= 2 \cdot \int \frac{ln ( \frac{s i n ( u )}{c o s ( u )} )}{sin ( 2 u )} d u

Use the basic trigonometric identity:

\frac{sin ( x )}{cos ( x )} = tan (x)

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

Rewrite using trig identities

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

Take the constant out:

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

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

Apply Trigonometric Substitution

= 2 \cdot \frac{1}{2} \cdot \int \frac{ln ( v )}{v} d v

Apply u-substitution

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

Apply the Power Rule

= 2 \cdot \frac{1}{2} \cdot \frac{w ^{2}}{2}

Substitute back

= 2 \cdot \frac{1}{2} \cdot \frac{ln ^{2} ( tan ( \frac{x}{2} ) )}{2}

Simplify

2 \cdot \frac{1}{2} \cdot \frac{ln ^{2} ( tan ( \frac{x}{2} ) )}{2} : \frac{1}{2} ln^{2} (tan (\frac{x}{2}))

= \frac{1}{2} ln^{2} (tan (\frac{x}{2}))

Add a constant to the solution

= \frac{1}{2} ln^{2} (tan (\frac{x}{2})) + C

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