Integrals are fundamental to calculus and are used to calculate areas, volumes, central points, and more. They come in two main types:

1. Definite Integrals: These compute the accumulation of quantities, typically over a specific interval $[a, b]$. The result is a number, representing, for instance, the area under a curve between the limits $a$ and $b$: $\int_a^b f(x) \, dx$

2. Indefinite Integrals: These find the antiderivative of a function. The result includes a constant of integration, $C$, since differentiation removes constants: $\int f(x) \, dx = F(x) + C$

Basic Integration Rules

1. Power Rule: $\int x^n \, dx = \frac{x^{n+1}}{n+1} + C, \quad n \neq -1$
2. Constant Multiple Rule: $\int k f(x) \, dx = k \int f(x) \, dx$
3. Sum Rule: $\int \big(f(x) + g(x)\big) \, dx = \int f(x) \, dx + \int g(x) \, dx$

Common Integral Results

1. $\int e^x \, dx = e^x + C$
2. $\int \sin(x) \, dx = -\cos(x) + C$
3. $\int \cos(x) \, dx = \sin(x) + C$
4. $\int \frac{1}{x} \, dx = \ln|x| + C, \, x \neq 0$

Applications of Integrals

1. Calculating Areas: Area under a curve $y = f(x)$ from $x = a$ to $x = b$.
2. Physics: Computing displacement, work, and energy in terms of varying forces or velocities.
3. Probability: Used in determining probabilities for continuous random variables.

Would you like examples, detailed explanations, or applications in a particular context?

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Relative Questions:

1. What is the difference between definite and indefinite integrals?
2. How do you compute integrals with substitution?
3. Can you explain integration by parts and when to use it?
4. What are improper integrals, and how are they evaluated?
5. How are integrals applied in real-world problems?

Tip: Always verify the domain of the function you're integrating, especially for definite integrals involving logarithms or rational functions.