Evaluate the following indefinite integral

Question

∫(2^√x)/(√x) dx

Answer 1

∫(2^√x)/(√x) dx

 

Note that √x = x^1/2.  Now let u = x^1/2:

• du = (1/2)x^-1/2 dx = (1/2u) dx.
• Hence 2u du = dx. 

 

Now substitute:

 

      ∫(2^u/u) 2u du = 2 ∫ 2^u du

 

Note that 2^u = e^{ln(2^u)} = e^uln(2)

 

      2 ∫ e^uln(2) du = (2/ln(2))*e^uln(2) + C =  (2/ln(2))*2u + C = (2/ln(2))*2^√x + C

 

Where C is the constant of integration.