1. **Stating the problem:** You asked about the antiderivative, integration, their graphical representation, real-life examples, and important formulas of integration. 2. **What is an antiderivative?** The antiderivative of a function $f(x)$ is another function $F(x)$ such that $$F'(x) = f(x).$$ It is also called the indefinite integral. 3. **What is integration?** Integration is the process of finding the antiderivative. It can be written as $$\int f(x)\,dx = F(x) + C,$$ where $C$ is the constant of integration. 4. **Graphical representation:** The graph of the antiderivative $F(x)$ is a curve whose slope at any point $x$ equals the value of $f(x)$ at that point. 5. **Real-life examples:** - If $f(x)$ represents velocity, then $F(x)$ represents position. - If $f(x)$ is the rate of water flow into a tank, $F(x)$ is the total volume of water. - If $f(x)$ is the rate of change of temperature, $F(x)$ is the temperature itself. 6. **Important formulas of integration:** - Power rule: $$\int x^n\,dx = \frac{x^{n+1}}{n+1} + C, \quad n \neq -1$$ - Constant multiple rule: $$\int a f(x)\,dx = a \int f(x)\,dx$$ - Sum rule: $$\int (f(x) + g(x))\,dx = \int f(x)\,dx + \int g(x)\,dx$$ - Integral of exponential: $$\int e^x\,dx = e^x + C$$ - Integral of sine and cosine: $$\int \sin x\,dx = -\cos x + C, \quad \int \cos x\,dx = \sin x + C$$