Is my prof crazy? Antiderivative of e^sqrt(x) without integration by parts?

Question

Is it possible to integrate something like e^sqrt(x) without using integration by parts or anything but substitution really?

Answer 1

You cannot integrate by parts. You can never do by parts nor ever by substitution.

You can integrate numerically by the Euler's method.
₢

Answer 2

Your proof is not crazy............substitute sqrt(x) = z
Then,
∫ e^{sqrt(x)} dx
= ∫ 2ze^(z) dz
= 2*∫ ze^(z) dz
= 2*∫ [d{ze^(z)} - e^(z)dz]
= 2ze^(z) - 2∫ e^(z)dz
= 2ze^(z) - 2e^(z) + c
= 2[e^{sqrt(x)}][sqrt(x) - 1] + c