Question

Evaluate the integral, if it exists.

\(\int {\frac{{e\sqrt x }}{{\sqrt x }}} dx\)

Short answer

The integral\(\int {\frac{{e\sqrt x }}{{\sqrt x }}} dx\) exists and the value for the integral is obtained as \(I = 2{e^{\sqrt x }} + c\).

Step-by-step solution

Integral definition

In mathematics, an integral is a numerical number equal to the area under a function's graph for some interval or a new function whose derivative equals the original function.

Evaluation of the integral

Evaluate the integral –

\(I = \int {\frac{{e\sqrt x }}{{\sqrt x }}} dx\)

Now, consider that –

\(\begin{aligned}{c}u &= \sqrt x \\du &= \frac{1}{{2\sqrt x }}dx\\dx &= 2\sqrt x du\end{aligned}\)

Substitute out the \(x\) and \(dx\) for \(u\) and \(du\) -

\(I = 2\int {{e^u}du} \)

Substitution and further calculation

It is known that \(\int {{e^x}dx = {e^x} + c} \), so –

\(I = 2{e^u} + c\)

Substitute back \(u = \sqrt x \)-

\(I = 2{e^{\sqrt x }} + c\)

Therefore, the value of the integral is \(I = 2{e^{\sqrt x }} + c\).