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Indefinite Integrals quiz #1

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  • What is an indefinite integral and how is it related to the antiderivative?
    An indefinite integral is the reverse process of differentiation, representing the antiderivative of a function. It is denoted by the integral symbol and always includes a constant of integration, c.
  • What does the notation ∫f(x) dx represent in calculus?
    The notation ∫f(x) dx represents the indefinite integral of the function f(x) with respect to x, meaning the antiderivative of f(x) plus a constant of integration.
  • What is the 'constant of integration' and why is it included in indefinite integrals?
    The constant of integration, denoted as c, is included because integrating a function can yield any function that differs by a constant, since the derivative of a constant is zero.
  • State the power rule for indefinite integrals and any restriction on its use.
    The power rule for indefinite integrals is ∫xⁿ dx = xⁿ⁺¹/(n+1) + c, where n ≠ -1. The restriction is that n cannot be -1 because division by zero is undefined.
  • How do you integrate a constant, such as ∫a dx?
    To integrate a constant a, use ∫a dx = a·x + c.
  • What is the indefinite integral of 0 with respect to x?
    The indefinite integral of 0 with respect to x is a constant, c.
  • How do the sum and difference rules apply to indefinite integrals?
    The sum and difference rules state that the integral of a sum or difference of functions equals the sum or difference of their integrals: ∫[f(x) ± g(x)] dx = ∫f(x) dx ± ∫g(x) dx.
  • What is the constant multiple rule for indefinite integrals?
    The constant multiple rule states that a constant can be factored out of the integral: ∫a·f(x) dx = a·∫f(x) dx.
  • Find the indefinite integral of 3x² with respect to x.
    ∫3x² dx = x³ + c.
  • Find the indefinite integral of x⁶ with respect to x.
    ∫x⁶ dx = x⁷/7 + c.
  • Find the indefinite integral of t⁴ with respect to t.
    ∫t⁴ dt = t⁵/5 + c.
  • Find the indefinite integral of 5x³ with respect to x.
    ∫5x³ dx = (5/4)x⁴ + c.
  • Find the indefinite integral of x² - 3x + 6 with respect to x.
    ∫(x² - 3x + 6) dx = (1/3)x³ - (3/2)x² + 6x + c.