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

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

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

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

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

Solution steps

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

Apply u-substitution

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

Apply u-substitution

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

Use the common integral:

\int cos (v) d v = sin (v)

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

Substitute back

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

Add a constant to the solution

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

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