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Evaluate the integral... Se^(3x)cos(2x)dx S=Integral

Se^(3x)cos(2x)dx  S=Integral

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James R. | PhD Graduate for Physics, Chemistry, Mathematics TutoringPhD Graduate for Physics, Chemistry, Mat...
4.9 4.9 (88 lesson ratings) (88)
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The first step is to convert the cosine function to the exponential form using Euler's formula.
Cos(2x) = 1/2 * [e^(i2x)-e^(-i2x)]
The next step is to distribute the outside exponential that is 1/2*[e^(3x+i2x)-e^(3x-i2x)].
Now the integral becomes a very simple u and v substitution integral. That is let u = (3+i2)x and v = (3-i2)x therefore du = (3+i2)dx and dv = (3-i2)dx.
Is this far enough help or do you need the next steps?
Kirill Z. | Physics, math tutor with great knowledge and teaching skillsPhysics, math tutor with great knowledge...
4.9 4.9 (174 lesson ratings) (174)
You can do it by applying the integration by parts.
Now let I=∫e3xcos(2x)dx; Then we have an equation:
Solving for I yields:


I do get the e^(3x)/13 in front but I have a much more simple expression in the brackets.
I checked with Mathematica. My answer is correct. ;-)
After relooking at my method, it does simplify to the answer you give. It just gives the student two different methods for solving the same problem.