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kennis2
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Integral of e^(2x)sin[3x]??
Integration by parts of: e^(2x)sin[3x]
result:e^2x(2sin3x-3cos3x)/13+C
can't get that result =(
Integration by parts of: e^(2x)sin[3x]
result:e^2x(2sin3x-3cos3x)/13+C
can't get that result =(
dextercioby said:Here's a very elegant way
[tex] \int e^{2x}\sin 3x \ dx=\mbox{Im}\left(\int e^{(2+3i)x} \ dx\right)=\mbox{Im}\left(\frac{e^{2x+3ix}}{2+3i}\right)+C=...=\frac{2\sin 3x-3\cos 3x}{13}e^{2x}+C [/tex]
The formula for the integral of e^(2x)sin[3x] is ∫e^(2x)sin[3x]dx = (1/13)e^(2x)(3sin[3x]-2cos[3x]) + C.
To solve this integral using integration by parts, you would set u = sin[3x] and dv/dx = e^(2x). Then, integrate both sides to find v = (1/2)e^(2x), and use the formula ∫udv = uv - ∫vdu to solve for the integral.
Yes, you can use trigonometric substitution to solve this integral. You can substitute u = 3x and du = 3dx, which will change the integral to ∫e^(2/3)u*sin[u]du. Then, you can use the formula ∫e^(ax)sin[bx]dx = (1/a^2+b^2)e^(ax)(asin[bx]-bcos[bx]) + C to solve the integral.
It depends on the individual's level of understanding and familiarity with integration techniques. For someone who is experienced and well-versed in integration, this integral may not be considered difficult. However, for someone who is new to integration, this integral may pose more of a challenge.
This integral has various applications in physics, engineering, and economics. For example, it can be used to model the growth of a population or the decay of radioactive substances. It can also be used to calculate the work done by a force acting on a moving object or the power output of an electric circuit. In economics, it can be used to calculate the present value of a future cash flow.