For integrals involving powers of \( x \), use substitution to simplify the expression. In this case, two substitutions were required to reduce the integral to a basic form.
We are asked to solve the integral:
\[
I = \int \frac{\sqrt{x}}{1 + \sqrt{x^{3/2}}} \, dx
\]
Step 1: Simplify the expression inside the integral. Recall that:
\[
\sqrt{x} = x^{1/2} \quad \text{and} \quad \sqrt{x^{3/2}} = x^{3/4}
\]
Thus, the integral becomes:
\[
I = \int \frac{x^{1/2}}{1 + x^{3/4}} \, dx
\]
Step 2: Use substitution to simplify the integral. Let:
\[
u = x^{1/4} \quad \text{so that} \quad x = u^4 \quad \text{and} \quad dx = 4u^3 \, du
\]
Step 3: Substitute into the integral:
\[
I = \int \frac{(u^4)^{1/2}}{1 + u^3} \cdot 4u^3 \, du
\]
Simplifying:
\[
I = \int \frac{u^2}{1 + u^3} \cdot 4u^3 \, du
\]
\[
I = 4 \int \frac{u^5}{1 + u^3} \, du
\]
Step 4: Perform another substitution. Let:
\[
v = 1 + u^3 \quad \text{so that} \quad dv = 3u^2 \, du
\]
Step 5: Substitute into the integral:
\[
I = \frac{4}{3} \int \frac{v}{v} \, dv = \frac{4}{3} \int 1 \, dv
\]
Step 6: Integrate:
\[
I = \frac{4}{3} \cdot v + C
\]
Substitute back \( v = 1 + u^3 \) and \( u = x^{1/4} \):
\[
I = \frac{4}{3} \left( 1 + (x^{1/4})^3 \right) + C = \frac{4}{3} \left( 1 + x^{3/4} \right) + C
\]
Thus, the solution to the integral is:
\[
\boxed{I = \frac{4}{3} \left( 1 + x^{3/4} \right) + C}
\]
