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Integrate 2^(x)

  1. Dec 5, 2003 #1
    Could someone please help me with the integral of 2^x. dx

    I bet its really simple but i have looked in several books and they just give the answer.
  2. jcsd
  3. Dec 5, 2003 #2
    1.The simplest way to solve it is to remember what is the derivative of 2x,by integrating the known equality.

    (In the general case [ax]'=ax*lna with a=const)

    2.Let 2x=t

    x=(1/ln2)*lnt ---> dx=(1/ln2)*1/t*dt

    Further is straightforward.
    Last edited: Dec 5, 2003
  4. Dec 5, 2003 #3


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    One way to do this is to note that, since ex and ln(x) are inverse functions, x= eln(x) for all x.

    In particular, 2x= e^(ln(2x)= ex ln(2)

    so that d(2x)/dx= dex ln(2)/dx= ln(2) 2x. (I'll bet that derivative formula is somewhere in your text.)

    Since d(2x)/dx= ln(2) 2x,
    the anti-derivative of 2x is (1/ln(2)) 2x.

    In general, the derivative of ax is ln(a) ax and the anti-derivative is (1/ln(a)) ax.

    (Notice that if a= e, ln(e)= 1 and we get the standard formulas.)
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