Use the formula below or any other method for the integration of rational functions in Q1.
∫(ax+b)/(cx+d) dx=a/cx + (bc-ad)/c^2 ln│cx + d│+ c
Evaluate:
∫x^2/(x+1)  dx

To evaluate the integral \(\int \frac{x^2}{x+1} \, dx\), we will first simplify the expression using polynomial division, since the degree of the numerator is higher than the degree of the denominator.

### Step 1: Polynomial Division
We divide \(x^2\) by \(x+1\):

\[
\frac{x^2}{x+1} = x - 1 + \frac{1}{x+1}
\]

So, the integral becomes:
\[
\int \frac{x^2}{x+1} \, dx = \int \left( x - 1 + \frac{1}{x+1} \right) dx
\]

### Step 2: Break the integral into parts
Now we integrate each term separately:
\[
\int \left( x - 1 + \frac{1}{x+1} \right) dx = \int x \, dx - \int 1 \, dx + \int \frac{1}{x+1} \, dx
\]

### Step 3: Compute each integral
- \(\int x \, dx = \frac{x^2}{2}\)
- \(\int 1 \, dx = x\)
- \(\int \frac{1}{x+1} \, dx = \ln |x+1|\)

### Step 4: Combine the results
The full integral becomes:
\[
\frac{x^2}{2} - x + \ln |x+1| + C
\]

Thus, the solution to the integral is:
\[
\int \frac{x^2}{x+1} \, dx = \frac{x^2}{2} - x + \ln |x+1| + C
\]

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Would you like more details or have any questions?

Here are 5 related questions to extend your understanding:
1. How would the result change if the denominator was \(x+2\) instead of \(x+1\)?
2. Can you perform the integration of \(\int \frac{x^3}{x+1} \, dx\)?
3. What happens if you add a constant \(k\) to the numerator, such as \(x^2 + k\)?
4. Can you apply the same method to integrate \(\frac{x^2 + 2x + 1}{x+1} \, dx\)?
5. How would you integrate \(\frac{x}{x+1}\) directly without polynomial division?

**Tip:** Always check the degree of the numerator and denominator when integrating rational functions. If the degree of the numerator is larger, polynomial division can simplify the integral.

Use the formula below or any other method for the integration of rational functions in Q1. ∫(ax+b)/(cx+d) dx = a/cx + (bc-ad)/c^2 ln│cx + d│+ c Evaluate: ∫x^2/(x+1) dx

Learn how to evaluate the integral ∫x^2/(x+1) dx by simplifying the rational function using polynomial division. This step-by-step guide walks through each stage of the integration process, including handling logarithmic and linear terms. Suitable for advanced high school or undergraduate calculus students.

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