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Definition:
- Integration is the process of finding the integral of a function. It is the reverse operation of differentiation.
- The result of integration is called the antiderivative or indefinite integral of the function.
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Integral Notation:
- The integral of a function f(x) with respect to x is denoted by ∫f(x) dx.
- The symbol ∫ represents integration, f(x) is the integrand, and dx indicates the variable of integration.
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Definite Integral:
- A definite integral represents the area under the curve of a function between two specified limits of integration.
- It is denoted by ∫[a, b] f(x) dx, where a and b are the lower and upper limits of integration, respectively.
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Fundamental Theorem of Calculus:
- The Fundamental Theorem of Calculus establishes a connection between differentiation and integration.
- Part I states that if F(x) is an antiderivative of f(x), then ∫[a, b] f(x) dx = F(b) - F(a).
- Part II states that if f(x) is continuous on an interval [a, b], then F(x) = ∫[a, x] f(t) dt is an antiderivative of f(x).
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Integration Techniques:
- Substitution: Also known as the u-substitution method, it involves substituting a new variable to simplify the integrand.
- Integration by Parts: A technique based on the product rule for differentiation that allows us to integrate products of functions.
- Partial Fractions: Used to decompose rational functions into simpler fractions for integration.
- Trigonometric Integrals: Involves applying trigonometric identities to integrate trigonometric functions.
- Improper Integrals: Integrals with infinite limits or integrals with discontinuous integrands.
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Applications of Integration:
- Integration is used to find areas, volumes, arc lengths, surface areas, and various physical quantities in real-world applications.
- It is essential in physics, engineering, economics, and other fields for solving optimization problems and modeling continuous processes.
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