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Integration: Techniques & Applications

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Integration techniques and applications like area under a curve and volume.

calculusintegration

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Integral

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  1. 1Integral
    The continuous analog of a sum, used to calculate areas, volumes, and their generalizations; the inverse operation of differentiation
  2. 2Integration
    The process of computing an integral, one of the two fundamental operations of calculus (along with differentiation)
  3. 3Definite integral
    An integral with specified limits that computes the signed area of a region bounded by a function's graph between two points
  4. 4Indefinite integral
    An integral without limits that represents a class of antiderivative functions whose derivative is the integrand
  5. 5Antiderivative
    A function F(x) whose derivative is the given function f(x); that is, F'(x) = f(x)
  6. 6the integral symbol ∫
    Integration, introduced by Leibniz in 1675, adapted from the long s (ſ) standing for 'summa' (sum)
  7. 7What does dx mean in an integral?
    The differential of variable x, indicating that the variable of integration is x
  8. 8Integrand
    The function f(x) being integrated in ∫f(x)dx
  9. 9Limits of integration
    The points a and b in ∫[a to b] f(x)dx, defining the interval over which integration occurs
  10. 10What does the fundamental theorem of calculus state?
    Differentiation and integration are inverse operations; it provides a method to compute definite integrals using antiderivatives
  11. 11State the fundamental theorem of calculus (part 1)
    If F is an antiderivative of continuous f on [a,b], then ∫[a to b] f(x)dx = F(b) - F(a)
  12. 12State the fundamental theorem of calculus (part 2)
    If F(x) = ∫[a to x] f(t)dt and f is continuous, then F'(x) = f(x)
  13. 13Power rule for integration
    ∫x^n dx = x^(n+1)/(n+1) + C, for n ≠ -1
  14. 14Find ∫x³ dx
    x⁴/4 + C
  15. 15Find ∫x⁵ dx
    x⁶/6 + C
  16. 16Integral of 1/x
    ∫(1/x)dx = ln|x| + C, for x ≠ 0
  17. 17Why is there a +C in indefinite integrals?
    Because antiderivatives differ by a constant (the derivative of a constant is zero), so we add an arbitrary constant C
  18. 18Constant multiple rule for integrals
    ∫k·f(x)dx = k·∫f(x)dx, where k is a constant
  19. 19Sum rule for integrals
    ∫[f(x) + g(x)]dx = ∫f(x)dx + ∫g(x)dx
  20. 20Difference rule for integrals
    ∫[f(x) - g(x)]dx = ∫f(x)dx - ∫g(x)dx
  21. 21Find ∫(3x² + 5x - 2)dx
    x³ + (5/2)x² - 2x + C
  22. 22Integral of e^x
    ∫e^x dx = e^x + C
  23. 23Integral of a^x
    ∫a^x dx = a^x/ln(a) + C, for a > 0, a ≠ 1
  24. 24Integral of sin(x)
    ∫sin(x)dx = -cos(x) + C
  25. 25Integral of cos(x)
    ∫cos(x)dx = sin(x) + C
  26. 26Integral of sec²(x)
    ∫sec²(x)dx = tan(x) + C
  27. 27Integral of csc²(x)
    ∫csc²(x)dx = -cot(x) + C
  28. 28Integral of sec(x)tan(x)
    ∫sec(x)tan(x)dx = sec(x) + C
  29. 29Integral of csc(x)cot(x)
    ∫csc(x)cot(x)dx = -csc(x) + C
  30. 30Integral of 1/(1 + x²)
    ∫1/(1 + x²)dx = arctan(x) + C

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