# integrate 2^(x)

by MathematicalPhysics
Tags: integrate
 P: 40 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.
 P: 261 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.
 Math Emeritus Sci Advisor Thanks PF Gold P: 38,900 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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