To solve the integral

$\int \left(\frac{2x^3 - 3x^2 + 4x - 5}{x^2}\right) \, dx,$

let's first simplify the integrand by dividing each term in the numerator by $x^2$:

$\frac{2x^3}{x^2} - \frac{3x^2}{x^2} + \frac{4x}{x^2} - \frac{5}{x^2} = 2x - 3 + \frac{4}{x} - \frac{5}{x^2}.$

So the integral becomes:

$\int \left(2x - 3 + \frac{4}{x} - \frac{5}{x^2}\right) \, dx.$

Now, we can integrate each term separately:

1. $\int 2x \, dx = x^2$,
2. $\int -3 \, dx = -3x$,
3. $\int \frac{4}{x} \, dx = 4 \ln |x|$,
4. $\int -\frac{5}{x^2} \, dx = \frac{5}{x}$.

So the integral is:

$x^2 - 3x + 4 \ln |x| + \frac{5}{x} + C,$

where $C$ is the constant of integration.

Final Answer:

$\int \left(\frac{2x^3 - 3x^2 + 4x - 5}{x^2}\right) \, dx = x^2 - 3x + 4 \ln |x| + \frac{5}{x} + C.$

Would you like more details on any step or have any questions?

Here are 5 related questions for further practice:

1. How would you integrate $\int (x^3 - x^2 + 1) \, dx$?
2. What is the integral of $\int \left(\frac{x^4 - 2x^3 + x}{x^2}\right) \, dx$?
3. How do you handle the integration of a rational function like $\int \frac{3x^2 + 2x + 1}{x} \, dx$?
4. How do logarithmic functions like $\ln |x|$ appear in integrals?
5. What is the importance of simplifying the integrand before integrating?

Tip: Always simplify the integrand if possible before integrating, as it makes the process much easier and reduces the chances of making errors.