Find a general solution for $\int at^ne^{bt}dt$, where $n$ is any integer, and $a$ and $b$ are real constants.

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To find the general solution you can use integration by parts. The formula for integration by parts is given by:

$\int u \, dv = uv - \int v \, du $

Let's choose u and dv such that we can apply this formula. In this case, we can choose:

$u = t^n \quad \text{and} \quad dv = ae^{bt} \, dt $

Now, differentiate u to find du and integrate dv to find v:

$du = nt^{n-1} \, dt \quad \text{and} \quad v = \frac{a}{b}e^{bt}$

Now, apply the integration by parts formula:

$\int at^n e^{bt} \, dt = t^n \left(\frac{a}{b}e^{bt}\right) - \int \frac{a}{b}e^{bt} \cdot nt^{n-1} \, dt$

Simplify the result:

$= \frac{a}{b}t^n e^{bt} - \frac{an}{b} \int t^{n-1} e^{bt} \, dt$