Calculus Integration Study Notes

Calculus Study Notes

• Integration is the reverse process of differentiation.

• The symbol ∫ represents integration.

• dx indicates integration with respect to x; dy indicates integration with respect to y.

• There's no general rule for integrating fractions, roots, or products of functions.

• Indefinite Integrals:

  • The integral of a constant 'a' with respect to x is ax + c, where 'c' is the constant of integration.
  • The integral of xⁿ with respect to x is (xⁿ⁺¹) / (n + 1) + c.

• Examples of Indefinite Integrals:

  • ∫ 6 dx = 6x + c
  • ∫ √5 dx = √5x + c
  • ∫x²dx = (x³) / 3 + c

• Definite Integrals: The definite integral of a function f(x) from a to b is evaluated as [F(x)]_a^b = F(b) - F(a), where F(x) is the antiderivative of f(x). No constant of integration 'c' is added.

• Properties of Integrals:

  • The integral of a constant multiple of a function is the constant multiple of the integral of the function.
  • The integral of a sum or difference of functions is the sum or difference of their individual integrals.

• Rules for integrating functions with exponents.

• Rules for integrating functions with radicals.

• Rules for integrating polynomial functions.

• Integration rules and properties for different types of functions (e.g., exponential, trigonometric, logarithmic, and hyperbolic).

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