The integral shown in your image is:

$\int \frac{x^2 + 2}{x^2 - 3} \, dx$

Let me solve it step-by-step:

Step 1: Simplify the fraction

The numerator $x^2 + 2$ has a degree equal to the denominator $x^2 - 3$. Therefore, we perform polynomial long division.

Divide $x^2 + 2$ by $x^2 - 3$:

• The quotient is $1$.
• The remainder is $(2 + 3) = 5$.

Thus, we can rewrite the integrand as: $\frac{x^2 + 2}{x^2 - 3} = 1 + \frac{5}{x^2 - 3}.$

Step 2: Break the integral into parts

Split the integral: $\int \frac{x^2 + 2}{x^2 - 3} \, dx = \int 1 \, dx + \int \frac{5}{x^2 - 3} \, dx.$

Step 3: Solve each part

1. The first term: $\int 1 \, dx = x.$

2. The second term: For $\int \frac{5}{x^2 - 3} \, dx$, let $a^2 = 3$, so the denominator becomes $(x^2 - a^2)$. Using the standard formula for this type of integral: $\int \frac{1}{x^2 - a^2} \, dx = \frac{1}{2a} \ln \left| \frac{x - a}{x + a} \right| + C,$ we compute: $\int \frac{5}{x^2 - 3} \, dx = \frac{5}{2\sqrt{3}} \ln \left| \frac{x - \sqrt{3}}{x + \sqrt{3}} \right|.$

Step 4: Combine the results

The final solution is: $\int \frac{x^2 + 2}{x^2 - 3} \, dx = x + \frac{5}{2\sqrt{3}} \ln \left| \frac{x - \sqrt{3}}{x + \sqrt{3}} \right| + C.$

Would you like a more detailed breakdown of any step?

Related Questions:

1. How do you perform polynomial long division for integrals?
2. Why do we rewrite $\frac{x^2 + 2}{x^2 - 3}$ as $1 + \frac{5}{x^2 - 3}$?
3. What is the derivation of the integral formula for $\int \frac{1}{x^2 - a^2} \, dx$?
4. How does the constant of integration $C$ affect the result?
5. Can this method be extended to higher-degree polynomials?

Tip:

Always check if the numerator's degree is greater than or equal to the denominator's degree before integrating rational functions. This helps decide if polynomial long division is required.