The problem is to evaluate the integral of a rational function: $\int \frac{1+3x+7x^2-2x^3}{x^2} dx$

1. Problem Description

The problem is to evaluate the integral of a rational function:

$\int \frac{1+3x+7x^2-2x^3}{x^2} dx$

2. Solution Steps

First, we can separate the fraction into individual terms:

$\int \frac{1+3x+7x^2-2x^3}{x^2} dx = \int (\frac{1}{x^2} + \frac{3x}{x^2} + \frac{7x^2}{x^2} - \frac{2x^3}{x^2}) dx$

Simplify the expression:

$= \int (x^{-2} + 3x^{-1} + 7 - 2x) dx$

Now, we integrate each term separately using the power rule $\int x^n dx = \frac{x^{n+1}}{n+1} + C$ for $n \neq -1$ and $\int \frac{1}{x} dx = \ln|x| + C$.

$\int x^{-2} dx = \frac{x^{-2+1}}{-2+1} = \frac{x^{-1}}{-1} = -x^{-1} = -\frac{1}{x}$

$\int 3x^{-1} dx = 3\int \frac{1}{x} dx = 3 \ln|x|$

$\int 7 dx = 7x$

$\int -2x dx = -2 \int x dx = -2 \frac{x^{1+1}}{1+1} = -2 \frac{x^2}{2} = -x^2$

Combining all the terms, we have:

$\int (x^{-2} + 3x^{-1} + 7 - 2x) dx = -\frac{1}{x} + 3 \ln|x| + 7x - x^2 + C$

3. Final Answer

The integral is:

$-\frac{1}{x} + 3\ln|x| + 7x - x^2 + C$