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

The integral of sin(e^{3x}) is 1/3 \Si(e^{3x})+C

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integral of sin(e^{3x})

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

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

Solution steps

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

Apply u-substitution

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

= \int sin (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 sin (e^{u}) d u

Apply u-substitution

: \int \frac{sin ( v )}{v} d v

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

Use the nonelementary integral:

\int \frac{sin ( x )}{x} d x = Si (x)

= \frac{1}{3} Si (v)

Substitute back

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

Add a constant to the solution

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

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