Evaluate the following integral:

∫
2
x
3
−
3
x
2
+
4
x
−
5
x
2
 
d
x
∫ 
x 
2
 
2x 
3
 −3x 
2
 +4x−5
​
 dx

Hint:
Try simplifying the integrand before integrating.​

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.

Learn how to evaluate the integral of (2x^3 - 3x^2 + 4x - 5) / x^2 with a detailed step-by-step solution. This problem covers integration techniques including rational functions and logarithmic functions. Suitable for advanced high school students and college-level calculus enthusiasts.

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