How to determine the indefinite integral of e^square rootx/square rootx (substitution or parts) ?

Expert Answers
justaguide eNotes educator| Certified Educator

We have to find the integral of e^ sqrt x / sqrt x.

Let u = sqrt x

=> du/dx = 1/2* sqrt x

=> 2* du = dx / sqrt x

Int [ (e^ sqrt x / sqrt x) dx]

=> Int [ e^u * (dx/sqrt x)]

=> Int [ e^u * 2 du]

=> 2*e^u + C

replace u = sqrt x

=> 2e^ sqrt x + C

Therefore the required integral is 2*e^ sqrt x + C

giorgiana1976 | Student

We'll integrate using substitution or we'll change the variable. For this reason, we'll consider the formula:

We'll note sqrtx = t.

We'll differentiate both sides:

dx/2sqrtx = dt

dx/sqrtx = 2dt

We'll re-write the integral in t:

Int e^t*2dt = 2e^t + C

We'll substitute t by sqrt x:

Int e^sqrtx*dx/sqrtx = 2e^sqrtx + C

The indefinite integral is evaluated using substitution and the result is:

Int e^sqrtx*dx/sqrtx = 2e^sqrtx + C