Integral Evaluation for \(\smallint \frac{1}{{x + \sqrt x }}\)dx

We want to find the value of the integral given by: \(\smallint \frac{1}{{x + \sqrt x }}\)dx

First, let's simplify the denominator of the integrand. We can factor out \(\sqrt x\) from the terms in the denominator:

\(x + \sqrt x = \sqrt x \cdot \sqrt x + \sqrt x = \sqrt x (\sqrt x + 1)\)

So the integral becomes:

\(\smallint \frac{1}{{\sqrt x (\sqrt x + 1)}}\)dx

Substitution Method for Integration

This form suggests using a substitution to make the integration easier. Let's choose a substitution involving the term \(\sqrt x + 1\).

Let \(u = \sqrt x + 1\).

Now, we need to find the differential \(du\) in terms of \(dx\). We differentiate \(u\) with respect to \(x\):

\(\frac{du}{dx} = \frac{d}{dx}(\sqrt x + 1)\)

Recall that \(\sqrt x = x^{1/2}\). So, \(\frac{d}{dx}(\sqrt x) = \frac{1}{2}x^{(1/2 - 1)} = \frac{1}{2}x^{-1/2} = \frac{1}{2\sqrt x}\).

And \(\frac{d}{dx}(1) = 0\).

So, \(\frac{du}{dx} = \frac{1}{2\sqrt x}\).

Rearranging this to find \(dx\):

\(du = \frac{1}{2\sqrt x} dx\)

\(dx = 2\sqrt x du\)

Now, we substitute \(u\) and \(dx\) into the integral:

\(\smallint \frac{1}{{\sqrt x (\sqrt x + 1)}}\)dx = \(\smallint \frac{1}{{\sqrt x (u)}} (2\sqrt x du)\)

Notice that \(\sqrt x\) in the numerator from \(dx\) and \(\sqrt x\) in the denominator cancel out:

= \(\smallint \frac{2}{u}\)du

Evaluating the Substituted Integral

The integral is now much simpler:

\(\smallint \frac{2}{u}\)du

We can pull the constant 2 out of the integral:

= \(2 \smallint \frac{1}{u}\)du

The integral of \(\frac{1}{u}\) with respect to \(u\) is \(\log|u|\). So:

= \(2 \log|u| + C\)

(where C is the constant of integration)

Substituting Back to Find the Final Answer

Finally, we substitute back \(u = \sqrt x + 1\) into the expression:

= \(2 \log|\sqrt x + 1| + C\)

Since \(\sqrt x\) is always non-negative for real \(x\), \(\sqrt x + 1\) is always positive (assuming \(x \ge 0\) for the square root to be real). Therefore, the absolute value is not necessary.

The final value of the integral is \(2 \log(\sqrt x + 1) + C\).

Comparing this with the given options, we see that this matches option 3.