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I'm working on an ODE and ran into this integration by parts. My calculus is terrible. Can someone help?

[tex] e^{2t}x = \int e^{2t}cos(t) \ dt = \frac{1}{2}cos(t) \ e^{2t} + \frac{1}{2}\int e^{2t}sin(t)\ dt = \frac{1}{2}cos(t)\ e^{2t} + \frac{1}{2}\left(-e^{2t}cos(t) + 2\int cos(t)\ e^{2t} \right) = \frac{1}{2}cos(t) \e^{2t} - \frac{1}{2}cos(t)\ e^{2t} + 2\int cos(t)\ e^{2t} dt [/tex]

Now that can't be right. Where did I go wrong?

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# Integration by parts; the error of my ways

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