How to integrate 2^x?

Integration is the inverse process of differentiation.

Answer: The final value of the integration of  2^x is equal to 2^x/ln2 + c.

Go through the step-by-step solution to understand the explanation completely.

Explanation:

To find:  ∫ 2^xdx

We may make use of the log property, which states:

e^ln2 = 2

 ∫ 2^xdx = ∫ (e^ln2)^xdx

 ∫ 2^xdx =  ∫ (e^xln2)dx ------(1)

let u = xln2,

⇒ du/dx = ln2

⇒ dx = du/ln2 -----(2)

Using the value obtained from equation (2) in equation (1), we get:

∫ (e^xln2)dx = ∫ e^u×du/ln2

∫ (e^xln2)dx = 1/ln2 ∫ e^udu = 1/ln2 × e^u + c

Now putting back e^u = e^xln2 = 2^x we get,

1/ln2 × e^u + c = 1/ln2 × e^xln2 + c = 2^x/ln2 + c

Thus, the final value of the integration of  2^x gives the answer as  2^x/ln2 + c.