What is the integral of e^{3x}cot(e^{3x}) ?

The integral of e^{3x}cot(e^{3x}) is 1/3 ln|sin(e^{3x})|+C

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integral of e^{3x}cot(e^{3x})

\int e^{3 x} cot (e^{3 x}) d x

\frac{1}{3} ln sin (e^{3 x}) + C

Solution steps

\int e^{3 x} cot (e^{3 x}) d x

Apply u-substitution

: \int e^{u} cot (e^{u}) \frac{1}{3} d u

= \int e^{u} cot (e^{u}) \frac{1}{3} d u

Take the constant out:

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

= \frac{1}{3} \cdot \int e^{u} cot (e^{u}) d u

Apply u-substitution

: \int cot (v) d v

= \frac{1}{3} \cdot \int cot (v) d v

Rewrite using trig identities

= \frac{1}{3} \cdot \int \frac{cos ( v )}{sin ( v )} d v

Apply u-substitution

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

= \frac{1}{3} \cdot \int \frac{1}{w} d w

Use the common integral:

\int \frac{1}{w} d w = ln (∣ w ∣)

= \frac{1}{3} ln ∣ w ∣

Substitute back

= \frac{1}{3} ln sin (e^{3 x})

Add a constant to the solution

= \frac{1}{3} ln sin (e^{3 x}) + C

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What is the integral of e^{3x}cot(e^{3x}) ?
The integral of e^{3x}cot(e^{3x}) is 1/3 ln|sin(e^{3x})|+C