1. **State the problem:** We need to find the integral $$\int \left(\frac{x^{-4}+1}{x}\right) dx$$. 2. **Rewrite the integrand:** Simplify the expression inside the integral by dividing each term by $x$: $$\frac{x^{-4}}{x} + \frac{1}{x} = x^{-4-1} + x^{-1} = x^{-5} + x^{-1}$$ 3. **Integral formula:** Recall that $$\int x^n dx = \frac{x^{n+1}}{n+1} + C$$ for $n \neq -1$, and $$\int x^{-1} dx = \ln|x| + C$$. 4. **Apply the integral:** $$\int (x^{-5} + x^{-1}) dx = \int x^{-5} dx + \int x^{-1} dx$$ 5. **Calculate each integral:** - For $x^{-5}$, $n = -5$, so $$\int x^{-5} dx = \frac{x^{-5+1}}{-5+1} + C = \frac{x^{-4}}{-4} + C = -\frac{1}{4} x^{-4} + C$$ - For $x^{-1}$, $$\int x^{-1} dx = \ln|x| + C$$ 6. **Combine results:** $$\int \left(\frac{x^{-4}+1}{x}\right) dx = -\frac{1}{4} x^{-4} + \ln|x| + C$$ **Final answer:** $$\boxed{-\frac{1}{4} x^{-4} + \ln|x| + C}$$