Integrate 2^(x)

  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. 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. HallsofIvy

    HallsofIvy 40,306
    Staff Emeritus
    Science Advisor

    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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