Integrating a fraction generally involves finding the antiderivative of a function in the form of a fraction, which is \(\frac{f(x)}{g(x)}\). The approach to integrate a fraction depends on the specific form of the fraction. Here are some common methods:

### 1. **Direct Integration:**
   - If the fraction is simple, like \(\frac{k}{g(x)}\) where \(k\) is a constant, and \(g(x)\) is easily integrable, you can directly integrate.
   - Example: \(\int \frac{2}{x} dx = 2 \ln |x| + C\).

### 2. **Substitution Method:**
   - If the fraction can be simplified by a substitution, let \(u = g(x)\), then \(du = g'(x) dx\).
   - Example: \(\int \frac{1}{x^2 + 1} dx\) can be integrated by recognizing it as the derivative of \(\arctan(x)\), so the result is \(\arctan(x) + C\).

### 3. **Partial Fraction Decomposition:**
   - For rational functions where the degree of the numerator is less than the degree of the denominator, decompose the fraction into simpler fractions.
   - Example: \(\int \frac{3x + 1}{(x - 1)(x + 2)} dx\) can be decomposed into \(\int \left(\frac{A}{x - 1} + \frac{B}{x + 2}\right) dx\).

### 4. **Integration by Parts:**
   - If the fraction is a product of functions (e.g., \(f(x)g(x)\)), where \(f(x)\) is easier to differentiate and \(g(x)\) easier to integrate, you might use integration by parts.
   - Example: \(\int x e^x dx\) requires integration by parts.

### 5. **Trigonometric Substitution:**
   - Useful when dealing with integrals involving square roots or expressions like \(\frac{1}{a^2 + x^2}\).
   - Example: \(\int \frac{1}{\sqrt{a^2 - x^2}} dx\) involves the substitution \(x = a \sin(\theta)\).

### Example:
To integrate \(\frac{1}{x^2 + 4}\), you can use substitution or recognize it as a standard integral:
\[
\int \frac{1}{x^2 + 4} dx = \frac{1}{2} \arctan\left(\frac{x}{2}\right) + C
\]

Would you like more details on a specific method or another example?

### Related Questions:
1. How do you perform partial fraction decomposition?
2. What are some common substitutions used in integration?
3. How do you apply integration by parts to fractions?
4. Can you explain how trigonometric substitution works with an example?
5. How do you integrate improper fractions?

### Tip:
Always check if the fraction can be simplified before choosing a method of integration. Simplifying the expression first can make the integration process much easier.

Discover how to integrate a fraction step-by-step with methods like direct integration, substitution, partial fraction decomposition, integration by parts, and trigonometric substitution. Includes examples and tips for simplifying the integration process. Suitable for advanced high school and college students.

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