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    148361
  • 4.3 Properties of the integral

    Let f and g be two functions of x, and let c be a constant. We already know that for derivatives,

    eq_bb6ea640.png

    eq_e125f85d.png

    But since the indefinite integral is just the operation of undoing a derivative, the same kind of rules must hold true for indefinite integrals as well:

    eq_dbf9d0ff.png

    eq_4239b8f5.png

    And since a definite integral can be found by plugging in the upper and lower limits of integration into the indefinite integral, the same properties must be true of definite integrals as well.

    Example 5

    ◊ Evaluate the indefinite integral

    eq_3a9843bf.png

    ◊ Using the additive property, the integral becomes

    eq_eb23500d.png

    Then the property of scaling by a constant lets us change this to

    eq_13cc034b.png

    We need a function whose derivative is x, which would be x2/2, and one whose derivative is sin x, which must be -cos x, so the result is

    eq_fe822a70.png