Evaluate the integral of e^(2x) * cos(3x) dx.
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Step-by-Step Solution
Step 1: Apply Integration by Parts
We will use integration by parts to solve this integral. Let u=cos(3x) and dv=e2xdx. We then find du and $v.
Step 2: Calculate du and v
Differentiating u=cos(3x) gives du=−3sin(3x)dx. Integrating dv=e2xdx gives v=21e2x.
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, ∫e2xsin(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′=e2xdx. We find du′=3cos(3x)dx and v′=21e2x. Substituting these into the formula gives an expression that includes the original integral.
Step 5: Solve for the Original Integral
Let I=∫e2xcos(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.