1. **State the problem:** We want to express the integral $I(a,n+1)$ in terms of $I(a,n)$, where $I(a,n)$ is a function involving parameters $a$ and $n$. 2. **Assume the definition:** Typically, such integrals are defined as $$I(a,n) = \int x^n e^{ax} \, dx.$$ We want to find a relation for $$I(a,n+1) = \int x^{n+1} e^{ax} \, dx$$ in terms of $I(a,n)$. 3. **Use integration by parts:** Let - $u = x^{n+1}$, so $du = (n+1)x^n dx$ - $dv = e^{ax} dx$, so $v = \frac{e^{ax}}{a}$ 4. **Apply integration by parts formula:** $$I(a,n+1) = uv - \int v \, du = \frac{x^{n+1} e^{ax}}{a} - \int \frac{e^{ax}}{a} (n+1) x^n dx$$ 5. **Simplify the integral:** $$I(a,n+1) = \frac{x^{n+1} e^{ax}}{a} - \frac{n+1}{a} \int x^n e^{ax} dx = \frac{x^{n+1} e^{ax}}{a} - \frac{n+1}{a} I(a,n)$$ 6. **Final expression:** $$I(a,n+1) = \frac{x^{n+1} e^{ax}}{a} - \frac{n+1}{a} I(a,n) + C$$ where $C$ is the constant of integration. This formula expresses $I(a,n+1)$ in terms of $I(a,n)$ plus an explicit term involving $x^{n+1} e^{ax}$.