Evalutate the integral using integration by substitution or parts

Question

$$\int \frac{e^\sqrt{x}}{\sqrt{x}}dx$$

How do I calculate this using substitution or parts integration?

Answer 1

Just substitute $u = \sqrt x \implies du = \frac{1}{2\sqrt x} dx$, giving us: $$I = \int \frac{e^{\sqrt x}}{\sqrt x}\, dx = 2\int \frac{e^{\sqrt x}}{2\sqrt x}\, dx = 2\int e^u \, du = 2e^u + c = 2e^{\sqrt x} + c$$

Answer 2

Hint -

Put $\sqrt x = t$

$\frac 1{2\sqrt x} dx = dt$

$\frac{dx}{\sqrt x}= 2 dt$

Put above values and then,

I = $2 \int e^t dt$

Now integrate.

Answer 3

$$\int\frac{e^{\sqrt{x}}}{\sqrt{x}}dx=2\int d\left(e^{\sqrt{x}}\right)=2e^{\sqrt{x}}+C.$$