The conversation between human and AI assistant.
How can I solve: [math]\displaystyle\int \pi e^{2t}\cos\pi t+2e^{2t}\sin\pi t\;dt?[/math]
And what about [math]dv[/math]?
For [math]dv[/math], let's add up both [math]\pi e^{2t}[/math] and [math]2e^{2t}\sin\pi t[/math] together to get [math](\pi+2\sin\pi t)e^{2t}[/math]. So, for [math]dv[/math], we have [math]\pi e^{2t}+2e^{2t}\sin\pi t\, dt=(\pi+2\sin\pi t)e^{2t}\, dt[/math].