1. Let's start by stating the problem: Integration is the process of finding the integral of a function, which can be thought of as the area under the curve of that function. 2. The basic formula for integration is the reverse of differentiation. If $F(x)$ is the antiderivative of $f(x)$, then: $$\int f(x)\,dx = F(x) + C$$ where $C$ is the constant of integration. 3. Important rules to remember: - The integral of a sum is the sum of the integrals: $$\int (f(x) + g(x))\,dx = \int f(x)\,dx + \int g(x)\,dx$$ - The integral of a constant times a function is the constant times the integral of the function: $$\int a f(x)\,dx = a \int f(x)\,dx$$ 4. Example: Find $$\int 2x\,dx$$ - Using the power rule for integration: $$\int x^n\,dx = \frac{x^{n+1}}{n+1} + C$$ for $n \neq -1$. - Here, $n=1$, so: $$\int 2x\,dx = 2 \int x\,dx = 2 \cdot \frac{x^{1+1}}{1+1} + C = 2 \cdot \frac{x^2}{2} + C = x^2 + C$$ 5. So, the integral of $2x$ with respect to $x$ is: $$x^2 + C$$ Integration helps us find areas, solve differential equations, and analyze many physical phenomena.