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Math Help - antiderivative

  1. #1
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    antiderivative

    Hi I need some help with this antiderivative.

    cos(5x)+sinh(2x)
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  2. #2
    Senior Member
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    Antiderivative

    First, the antiderivative of a sum is the sum of the antiderivatives:
     \int \cos (5x) + \sinh (2x) dx = \int \cos (5x) dx + \int \sinh (2x) dx

    Then we look at \cos (5x). The antiderivative of \cos x is \sin x. By the chain rule

    \frac{d}{dx} \sin (5x) = \cos (5x) \cdot \frac{d}{dx} (5x) = 5 \cos (5x),
    thus \frac{d}{dx} (\tfrac{1}{5}\sin (5x)) = \tfrac{1}{5} \cdot 5 \cos (5x) = \cos (5x), and the antiderivative of \cos(5x) is \tfrac{1}{5}\sin (5x).

    Similar reasoning from the antiderivative of \sinh x will give you the other half of the antiderivative.

    --Kevin C.
    Last edited by TwistedOne151; December 6th 2007 at 10:30 AM. Reason: typo in formula
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