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.

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.

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.

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