Evaluate the integral of e^(2x) * cos(3x) dx.

Step 1: Apply Integration by Parts

We will use integration by parts to solve this integral. Let $u = \cos(3x)$ and $dv = e^{2x} \, dx$. We then find $du$ and v.

Step 2: Calculate $du$ and $v$

Differentiating $u = \cos(3x)$ gives $du = -3\sin(3x) \, dx$. Integrating $dv = e^{2x} \, dx$ gives $v = \frac{1}{2}e^{2x}$.

Step 3: Substitute into Integration by Parts Formula

Substitute the calculated values of $u, v, du, dv$ into the integration by parts formula. This results in a new integral, $\int e^{2x}\sin(3x) \, dx$, which also requires integration by parts.

Step 4: Apply Integration by Parts Again

For the new integral, we again apply integration by parts. Let $u' = \sin(3x)$ and $dv' = e^{2x} \, dx$. We find $du' = 3\cos(3x) \, dx$ and $v' = \frac{1}{2}e^{2x}$. Substituting these into the formula gives an expression that includes the original integral.

Step 5: Solve for the Original Integral

Let $I = \int e^{2x} \cos(3x) \, dx$. Substitute the result from the second integration by parts back into the equation from the first integration by parts. Then, rearrange the equation to solve for $I$. Finally, add the constant of integration C.

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