Integration, a cornerstone of calculus, is essentially the reverse of differentiation. It's a process used for finding the original function when its derivative is known. This section delves into the integration of functions in the form $(ax+b)^n$, where $n$ is a rational number, excluding -1. This covers the integration of constant multiples, sums, and differences of functions.

**Understanding Integration**

**Integration as the Reverse of Differentiation**:

This concept involves determining a function when its derivative is known, effectively reversing the process of differentiation.

**Notation and Terminology**:

- The integral sign, "∫", represents the operation of integration.
- The expression to be integrated, followed by "dx", indicates integration with respect to x.
- "dx" in integration corresponds to the "dx" in the derivative notation $\frac{dy}{dx}$.

**Types of Integrals**

**Indefinite Integrals**: These are integrals without specific limits, including a constant of integration, $+c$, to account for the loss of the original function's vertical position during differentiation.**Definite Integrals**: Integrals with specified upper and lower limits, used for calculating exact values, such as areas under curves. They exclude the constant $+c$.

**Integration Formulas**

- Basic Integral Form:

- Integral of a Function in the Form $(ax+b)^n$:

again, $n \neq -1.$

**Worked Examples**

**Example 1: **

Find the equation of a curve $y$ in terms of $x$ that passes through the point (1, 3), given$\frac{dy}{dx} = 6x^2$.

**Solution:**

**1. Integration**:

$y = \int 6x^2 \, dx = \frac{6x^{3}}{3} + c = 2x^3 + c.$

**2. Determining Constant **$c$:

Using the point (1, 3):

$3 = 2(1)^3 + c.$Solving for $c$:

$c = 1.$**3. Final Equation:**

Rahil spent ten years working as private tutor, teaching students for GCSEs, A-Levels, and university admissions. During his PhD he published papers on modelling infectious disease epidemics and was a tutor to undergraduate and masters students for mathematics courses.