Evaluate the integral. ∫ 28e^(√7x)/(2√x) dx

4√7 *e^√7x + c

To evaluate the integral ∫ 28e^(√7x)/(2√x) dx, we can use a technique called substitution.

Let's start by making a substitution u = √7x. To find the value of dx in terms of du, we differentiate both sides of the equation u = √7x with respect to x.

du/dx = √7

Rearranging the equation, we have dx = du/√7. Now, we can substitute these values into the integral and simplify it.

∫ 28e^(√7x)/(2√x) dx
= ∫ 28e^u/(2√(u^2/7)) * (du/√7)

Simplifying further, we get:

= 14/√7 ∫ e^u/√(u^2/7) du
= (14/√7) * ∫ e^u √(7/u^2) du
= 2√7 ∫ e^u/u du

Now, the integral has simplified to ∫ e^u/u du, which is known as the exponential integral Ei(u).

The exponential integral Ei(u) cannot be expressed in terms of elementary functions, but it can be approximated using numerical methods or computed using software such as calculators or computer algebra systems.

Therefore, the evaluation of the integral ∫ 28e^(√7x)/(2√x) dx involves computing the exponential integral Ei(u) for the value of u = √7x.