The given function is \( \displaystyle \frac{x^2 – 2x + 3}{x^4} \) and we need to find the integration of this function with respect to \( x \).
Solution:
\( \begin{align*} &\int \frac{x^2 – 2x + 3}{x^4} \, dx \\ &= \int \left( \frac{x^2}{x^4} \, – \frac{2x}{x^4} + \frac{3}{x^4} \right) dx \\ &= \int \left( x^{-2} – 2x^{-3} + 3x^{-4} \right) dx \\ &= \int x^{-2} \, dx \, – 2 \int x^{-3} \, dx + 3 \int x^{-4} \, dx \\ &= \frac{x^{-1}}{-1} \, – 2 \left( \frac{x^{-2}}{-2} \right) + 3 \left( \frac{x^{-3}}{-3} \right) + C \\ &= -x^{-1} + x^{-2} – x^{-3} + C, \quad \text{C is a constant} \\ \end{align*} \)Thus, the integration of the given function is
\( \boxed{\int \frac{x^2 – 2x + 3}{x^4} \, dx = -\frac{1}{x} + \frac{1}{x^2} \, – \frac{1}{x^3} + C } \\ \)Please let me know in the comments if you find any errors in this solution.