https://file.aiquickdraw.com/mathbot/file/172606168420992tft75v.jpg

Let's solve the integrals provided in Exercise 2.5 from the image:

### 1. \( \int \frac{x^4 + 5x^2 - 1}{x^2 - 9} \, dx \)

#### Step 1: Polynomial Long Division
- We divide \( x^4 + 5x^2 - 1 \) by \( x^2 - 9 \) using polynomial long division.

#### Step 2: Simplify and solve the integral
After division, the resulting expression is integrated term by term.

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### 2. \( \int \frac{5x - 11}{(x - 2)^2 (x + 1)^3} \, dx \)

#### Step 1: Partial Fraction Decomposition
- Express \( \frac{5x - 11}{(x - 2)^2 (x + 1)^3} \) in terms of partial fractions.

#### Step 2: Integrate each fraction
Once decomposed, integrate each term individually.

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### 3. \( \int \frac{x^2 + 7}{x^4 + 7x^2 + 12} \, dx \)

#### Step 1: Substitute \( u = x^2 \)
- Let \( u = x^2 \), which transforms the integral into a rational function in terms of \( u \).

#### Step 2: Solve the resulting integral
This transformed integral can be handled using partial fractions or another suitable method.

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### 4. \( \int \frac{2x + 1}{(x^2 + x + 1)^3} \,

Evaluate the following integrals: 1) ∫ (x^4 + 5x^2 - 1) / (x^2 - 9) dx, 2) ∫ (5x - 11) / [(x - 2)^2(x + 1)^3] dx, 3) ∫ (x^2 + 7) / (x^4 + 7x^2 + 12) dx, 4) ∫ (2x + 1) / (x^2 + x + 1)^3 dx.

This page provides step-by-step solutions for evaluating integrals using polynomial long division, partial fractions, and substitution methods. The problems involve rational functions, challenging algebraic manipulations, and calculus techniques, suitable for undergraduate mathematics students.

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