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Math Help - How do I find the antiderivative?

  1. #1
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    How do I find the antiderivative?

    I have a few homework problems, and they're asking me to find the antiderivative. They're quite tough, and I'm really confused. Can anyone be of some help on these few?

    integral(4x-5/x^3)dx (/ = over.)

    integral 3x over (x^2+4)dx. On this one, the 3x isn't in parentheses.

    integral 5xsec^2(x^2)dx

    integral (4x-5/x^2)dx

    integral e^4x dx

    integral sin(x)cos^3(x)dx

    Any help on ANY of these is greatly appreciated. I'm really lost
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  2. #2
    Senior Member Spec's Avatar
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    (1) \frac{3x}{x^2+4}
    We know that \frac{d}{dx}(\ln(x^2+4))= \frac{2x}{x^2+4}, so all we need to do is find a fractional number to multiply/divide \ln(x^2+4) with to get the antiderivative.

    (2) e^{4x}
    We know that \frac{d}{dx}(e^{4x})=4e^{4x}, so you simply need to multiply e^{4x} with \frac{1}{4} to get the antiderivative.

    (3) \sin(x)\cos^3(x)
    \frac{d}{dx}(\cos^4(x)) = 4\cos^3(x)\cdot -\sin(x)
    Again, look for the appropriate number to multiply/divide with.
    Last edited by Spec; April 29th 2009 at 09:14 AM. Reason: Misread first problem
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  3. #3
    Senior Member Twig's Avatar
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    hi

    hi

     \int (4x-\frac{5}{x^{3}}) \, dx = 2x^{2} +\frac{5}{2}x^{-2} + C \; ,\, C\in \mathbb{R}

     \int (\frac{3x}{x^{2}+4}) \, dx = \frac{3}{2}\cdot ln(x^{2}+4) + C

    Some of these are actually quite basic I am afraid.
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