Integration, the inverse operation to differentiation, is essential, enabling students to solve a broad spectrum of problems. This section focuses on extending basic integration methods to cover more complex functions such as $e^{ax+b}$, $\frac{1}{ax+b}$, $\sin(ax+b)$, $\cos(ax+b)$, $\sec^2(ax+b)$, and $\frac{1}{\sqrt{ax+b}}$.

Techniques for Advanced Integration

1. Integration of Exponential Functions

For an exponential function $e^{ax+b}$, the integral is

Example:

2. Integration of Rational Functions

Integrating a rational function $\frac{1}{ax+b}$, we find:

Example:

3. Integration of Trigonometric Functions

The integrals for trigonometric functions $\sin(ax+b)$ and $\cos(ax+b)$are:

Examples:

4. Integration of Secant Squared Function

The integral of $\sec^2(ax+b)$ is:

Example:

5. Integration of Radical Functions

For $\frac{1}{\sqrt{ax+b}}$, the correct integral involves an inverse sine function due to the square root in the denominator:

Example: