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Integrate 2^(x)

by MathematicalPhysics
Tags: integrate
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MathematicalPhysics
#1
Dec5-03, 03:33 AM
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.
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metacristi
#2
Dec5-03, 05:06 AM
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.
HallsofIvy
#3
Dec5-03, 07:16 AM
Math
Emeritus
Sci Advisor
Thanks
PF Gold
P: 39,490
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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