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

The integral of (csc^2(ln(x)))/x is ln|x|-ln(x)-cot(ln(x))+C

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integral of (csc^2(ln(x)))/x

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

ln ∣ x ∣ - ln (x) - cot (ln (x)) + C

Solution steps

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

Rewrite using trig identities

= \int \frac{1 + cot ^{2} ( ln ( x ) )}{x} d x

Expand

\frac{1 + cot ^{2} ( ln ( x ) )}{x} : \frac{1}{x} + \frac{cot ^{2} ( ln ( x ) )}{x}

= \int \frac{1}{x} + \frac{cot ^{2} ( ln ( x ) )}{x} d x

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

\int \frac{1}{x} d x = ln ∣ x ∣

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

= ln ∣ x ∣ - ln (x) - cot (ln (x))

Add a constant to the solution

= ln ∣ x ∣ - ln (x) - cot (ln (x)) + C

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What is the integral of (csc^2(ln(x)))/x ?
The integral of (csc^2(ln(x)))/x is ln|x|-ln(x)-cot(ln(x))+C