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

The integral of (cos(x))/(sin(x)ln(sin(x))) is ln|ln(sin(x))|+C

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

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

ln ∣ ln (sin (x)) ∣ + C

Solution steps

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

Use the basic trigonometric identity:

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

= \int \frac{cot ( x )}{ln ( sin ( x ) )} d x

Apply u-substitution

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

= \int \frac{1}{u} d u

Use the common integral:

\int \frac{1}{u} d u = ln (∣ u ∣)

= ln ∣ u ∣

Substitute back

u = ln (sin (x))

= ln ∣ ln (sin (x)) ∣

Add a constant to the solution

= ln ∣ ln (sin (x)) ∣ + C

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\int \frac{1}{30} d x

\int \frac{2 ( e ^{x} - 1 )}{e ^{x} - x} d x

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