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Antiderivatives

Antiderivatives. Indefinite Integral. The family of antiderivatives of a function f indicated by The symbol is a stylized S to indicate summation. Indefinite Integral. The indefinite integral is a family of functions The + C represents an arbitrary constant

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Antiderivatives

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  1. Antiderivatives

  2. Indefinite Integral • The family of antiderivatives of a function f indicated by • The symbol is a stylized S to indicate summation

  3. Indefinite Integral • The indefinite integral is a family of functions • The + C represents an arbitrary constant • The constant of integration

  4. Properties of Indefinite Integrals • The power rule • The integral of a sum (difference) is the sum (difference) of the integrals

  5. Properties of Indefinite Integrals • The derivative of the indefinite integral is the original function • A constant can be factored out of the integral

  6. Example : Evaluate

  7. Example : Find the function f such that • First find f (x) by integrating.

  8. Example : Evaluate and check by differentiation:

  9. Examples • Determine the indefinite integrals as specified below

  10. Integrate

  11. Find each antiderivative

  12. Find each antiderivative

  13. Find each antiderivative

  14. Find each antiderivative

  15. Solve the differential equation

  16. Solve the differential equation

  17. Solve the differential equation

  18. Solve the differential equation

  19. Given that the graph of f(x) passes through the point (1,6) and that the slope of its tangent line at (x.f(x) is 2x+1, find f(6)

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