What is the Integral of 2^(x)?

[MathematicalPhysics]

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.

 

[metacristi]

1.The simplest way to solve it is to remember what is the derivative of 2^x,by integrating the known equality.

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

2.Let 2^x=t

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

Further is straightforward.

 

[HallsofIvy]

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

In particular, 2^x= e^(ln(2^x)= e^{x ln(2)}

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

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

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

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