Integration is allowing us to find areas, volumes, and solve numerous real-world problems. This section focuses on integrating special functions, specifically $x^2 + ax$ and $\tan(x)$, which are prevalent in advanced mathematics. Understanding these integrations is crucial for solving complex calculus problems.

## Integration Techniques

### 1. Integration of $x^2 + ax$

The function $x^2 + ax$ represents a quadratic-linear form and is integrated by addressing each term individually.

- For the term $x^2$, the integral is $\frac{x^3}{3}$.
- For the term $ax$, the integral is $\frac{ax^2}{2}$. Combining these, we obtain: $\int (x^2 + ax) \, dx = \frac{x^3}{3} + \frac{ax^2}{2} + C$ where C represents the constant of integration.

### 2. Integration of $\tan(x)$

Integrating the trigonometric function $\tan(x)$ can be intricate. Utilising the identity $\tan(x) = \frac{\sin(x)}{\cos(x)}$, we approach the integral involving a logarithmic function: $\int \tan(x) \, dx = -\ln|\cos(x)| + C.$ This result is derived through substitution methods and leveraging the properties of logarithmic functions.

## Advanced Example Integrations

### Example 1: Integrating $x^2 + 3x$

**Identify the Function:**The function is $x^2 + 3x$.**Apply Integration Rules:**- Integrate $x^2$: Using $\int x^n \, dx = \frac{x^{n+1}}{n+1}$, we get $\int x^2 \, dx = \frac{x^3}{3}$.
- Integrate $3x$: Applying the power rule,$\int 3x \, dx = \frac{3x^2}{2}$.

**Combine Results:**Thus, $\int (x^2 + 3x) \, dx = \frac{x^3}{3} + \frac{3x^2}{2} + C$

### Example 2: Integrating $\tan(x)$

**Identify the Function:**The function is $\tan(x)$.**Apply Integration Rules:**The integral of $\tan(x)$ is known: $\int \tan(x) \, dx = -\ln|\cos(x)| + C$

### Example 3: Integrating $x^2 + 4x$

**Identify the Function:**The function is $x^2 + 4x$.**Apply Integration Rules:**- Integrate $x^2$: $\int x^2 \, dx = \frac{x^3}{3}$.
- Integrate $4x$: $\int 4x \, dx = 2x^2$.

**Combine Results:**Hence, $\int (x^2 + 4x) \, dx = \frac{x^3}{3} + 2x^2 + C$

### Example 4: Integrating $\tan(2x)$

**Identify the Function:**The function is $tan(2x)$.**Apply Integration Rules with a Substitution:**- Let $u = 2x$, then $du = 2dx$ or $dx = \frac{du}{2}$.
- Substitute into the integral: $\int \tan(2x) \, dx = \int \tan(u) \cdot \frac{du}{2}$
- Now integrate: $\int \tan(u) \, du = -\ln|\cos(u)|$.

**Back-Substitute and Combine Results:**- Replace $u$ with $2x: -\ln|\cos(2x)|$.
- Include the factor of $\frac{1}{2}$ from the substitution: $-\frac{1}{2} \ln|\cos(2x)| + C$.