The problem asks us to calculate the indefinite integral of the function $\frac{1+3x+7x^2-2x^3}{x^2}$ with respect to $x$.

1. Problem Description

The problem asks us to calculate the indefinite integral of the function $\frac{1+3x+7x^2-2x^3}{x^2}$ with respect to $x$.

2. Solution Steps

First, we simplify the expression by dividing each term in the numerator by $x^2$:

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

This simplifies to:

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

Which can also be written as:

$\int \left(x^{-2} + \frac{3}{x} + 7 - 2x\right) \, dx$

Now, we integrate each term separately:

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

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

$\int 7 \, dx = 7x$

$\int -2x \, dx = -2 \int x \, dx = -2 \cdot \frac{x^2}{2} = -x^2$

Combining these results and adding the constant of integration $C$, we get:

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

3. Final Answer

The final answer is $-\frac{1}{x} + 3\ln|x| + 7x - x^2 + C$.