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]
Okay, then what is [math]v[/math]?
We can integrate [math]v[/math] by replacing [math]e^{2t}[/math] with [math]u[/math] and [math]\sin\pi t[/math] with [math]v[/math]. 
Using integration by parts, we can have [math]\int e^{2t}\, dt = \frac{1}{2}e^{2t}[/math]. For [math]\int \sin\pi t\, dt[/math], we can let [math]w=\pi t[/math], therefore [math]\frac{dw}{dt}=\pi[/math], and [math]\int \sin\pi t\, dt=\int \frac{\sin w}{\pi}\, dw=-\frac{1}{\pi}\cos\pi t[/math]. Thus, [math]v=\frac{1}{2}e^{2t}-\frac{1}{\pi}\cos\pi t[/math].