91Ó°ÊÓ

Recall the formula for integration by parts: \(\int u \, dv = uv - \int v\, du\). By choosing appropriate functions for \(u\) and \(dv\), we aim to simplify the integration process.

In the given integral \(\int x^{n} e^{a x} d x\), we need to choose \(u\) and \(dv\) such that the resulting new integral is simpler. We can choose \(u = x^{n}\) and \(dv = e^{ax} dx\), because taking the derivative of \(x^n\) will simplify the expression, while taking the integral of \(e^{ax}\) remains manageable.

From our choice of \(u\) and \(dv\), we need to find their corresponding \(du\) and \(v\). Differentiate \(u\) with respect to \(x\): \(du = \frac{d}{dx} {x^n} = n x^{n-1} dx\) Integrate \(dv\) with respect to \(x\): \(v = \int e^{ax} dx = \frac{1}{a}e^{ax} + C\)

Now, we will apply the formula \(\int u \, dv = uv - \int v\, du\) using our expressions for \(u\), \(dv\), \(du\), and \(v\). By plugging in the values, we get: \(\int x^{n} e^{a x} d x = x^n (\frac{1}{a}e^{ax}) - \int (\frac{1}{a}e^{ax} + C)(n x^{n-1} dx) \) Note that when you integrate, the constant \(C\) can be ignored as it only affects the constant of integration in the final result, so we simplify to: \(\int x^{n} e^{a x} d x = x^n (\frac{1}{a}e^{ax}) - \frac{n}{a} \int x^{n-1} e^{a x} dx\) To further evaluate this integral, you might need to apply integration by parts again, depending on the value of \(n\). However, the main goal of the problem was to choose \(dv\) for this integral, which has been achieved.