Q. \(\int x^{1/2}\,\mathrm{d}x\)

We want to compute the indefinite integral

\[
\int x^{1/2}\,dx
\]

Step 1: Identify the power rule for integrals.

If \(n \neq -1\), then

\[
\int x^{n}\,dx = \frac{x^{n+1}}{n+1} + C
\]

Step 2: Match the given expression to the rule.

Here, the integrand is \(x^{1/2}\). So the exponent is

\[
n = \frac{1}{2}
\]

Step 3: Add \(1\) to the exponent.

\[
n+1 = \frac{1}{2} + 1 = \frac{1}{2} + \frac{2}{2} = \frac{3}{2}
\]

Step 4: Divide by the new exponent.

Using the rule, we get

\[
\int x^{1/2}\,dx = \frac{x^{3/2}}{\frac{3}{2}} + C
\]

Step 5: Simplify the coefficient.

Dividing by \(\frac{3}{2}\) is the same as multiplying by \(\frac{2}{3}\):

\[
\frac{1}{\frac{3}{2}} = \frac{2}{3}
\]

So the result becomes

\[
\int x^{1/2}\,dx = \frac{2}{3}x^{3/2} + C
\]

Final Answer:

\[
\boxed{\frac{2}{3}x^{3/2} + C}
\]