Integration of 2^x | 2^x Integration

The integration of 2^x is equal to 2^x/ln2 where ln denotes the natural logarithm, that is, ln=log_e. In this post, we will learn to integrate 2 to the power x with respect to x.

Integration of 2^x Formula

The integration formula of 2^x (2 to the x) with respect to x is given below.

∫ 2^x dx = 2^x/ln2 +C.

How to Integrate 2^x

Now, we will learn how to integrate the function 2 to the x and prove that ∫ 2^x dx = 2^x/ln2 +C.

Proof:

Note that x can be written as x=e^lnx, so we have

2=e^ln2 …(I)

Then, using this identity, the integration of 2^x will be equal to

∫2^x dx = ∫(e^ln2)^x dx

⇒ ∫2^x dx = ∫e^xln2 dx …(II)

Let xln2 =z. Then ln2 dx =dz

⇒ dx = dz/ln2

Now, from from (II) we get that

∫2^x dx = ∫e^z $\dfrac{dz}{\ln 2}$

= $\dfrac{1}{\ln 2}$ ∫e^z dz

= e^z/ln2 as the integration of e^z with respect to z is e^z.

= e^xln2/ln2 as z=xln2.

= 2^x/ln2 using the identity (I) as e^xln2 = (e^ln2)^x =(2)^x = 2^x.

So the integration of 2^x is equal to 2^x/ln2 where ln2=log_e2. This is obtained by the substitution method of integration and using the exponential identity x=e^lnx.

Video Solution of Integration of 2^x:

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