What is the integral of (cot^2(ln(x)))/(xcsc(ln(x))) ?

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

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integral of (cot^2(ln(x)))/(xcsc(ln(x)))

\int \frac{cot ^{2} ( ln ( x ) )}{x csc ( ln ( x ) )} d x

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

Solution steps

\int \frac{cot ^{2} ( ln ( x ) )}{x csc ( ln ( x ) )} d x

Apply u-substitution

= \int cot^{2} (u) sin (u) d u

Simplify

cot^{2} (u) sin (u) : cos (u) cot (u)

= \int cos (u) cot (u) d u

Rewrite using trig identities

= \int \frac{1 - sin ^{2} ( u )}{sin ( u )} d u

Expand

\frac{1 - sin ^{2} ( u )}{sin ( u )} : \frac{1}{sin ( u )} - sin (u)

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

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

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

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

\int sin (u) d u = - cos (u)

= ln tan (\frac{u}{2}) - (- cos (u))

Substitute back

u = ln (x)

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

Simplify

ln tan (\frac{ln ( x )}{2}) - (- cos (ln (x))) : ln tan (\frac{1}{2} ln (x)) + cos (ln (x))

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

Add a constant to the solution

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

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