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

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

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integral of (cos(e^x))/(sin(e^x))e^x

\int \frac{cos ( e ^{x} )}{sin ( e ^{x} )} e^{x} d x

ln ∣ sin (e^{x}) ∣ + C

Solution steps

\int \frac{cos ( e ^{x} )}{sin ( e ^{x} )} e^{x} d x

Simplify

\frac{cos ( e ^{x} )}{sin ( e ^{x} )} e^{x} : e^{x} cot (e^{x})

= \int e^{x} cot (e^{x}) d x

Apply u-substitution

: \int cot (u) d u

= \int cot (u) d u

Rewrite using trig identities

= \int \frac{cos ( u )}{sin ( u )} d u

Apply u-substitution

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

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

Use the common integral:

\int \frac{1}{v} d v = ln (∣ v ∣)

= ln ∣ v ∣

Substitute back

= ln ∣ sin (e^{x}) ∣

Add a constant to the solution

= ln ∣ sin (e^{x}) ∣ + C

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