To solve the integral \( \int \frac{\sqrt{x}}{\sqrt{a^3 - x^3}} \, dx \), we can follow these steps:
### Step 1: Rewrite the Integral
We start by rewriting the integral in a more manageable form:
\[
\int \frac{\sqrt{x}}{\sqrt{a^3 - x^3}} \, dx = \int \frac{\sqrt{x}}{(a^{3/2} - x^{3/2})} \, dx
\]
### Step 2: Substitution
Let us make the substitution \( x^{3/2} = t \). Then, differentiating both sides gives:
\[
\frac{3}{2} x^{1/2} \, dx = dt \quad \Rightarrow \quad dx = \frac{2}{3} \frac{dt}{\sqrt{x}} = \frac{2}{3} \frac{dt}{\sqrt{t^{2/3}}} = \frac{2}{3} \frac{dt}{t^{1/3}}
\]
### Step 3: Substitute in the Integral
Now substitute \( x^{3/2} = t \) into the integral:
\[
\int \frac{\sqrt{x}}{\sqrt{a^3 - x^3}} \, dx = \int \frac{\sqrt{t^{2/3}}}{\sqrt{a^3 - t}} \cdot \frac{2}{3} \frac{dt}{t^{1/3}} = \int \frac{2}{3} \frac{t^{1/3}}{\sqrt{a^3 - t}} \, dt
\]
### Step 4: Simplify the Integral
The integral now becomes:
\[
\frac{2}{3} \int \frac{t^{1/3}}{\sqrt{a^3 - t}} \, dt
\]
### Step 5: Use a Standard Integral Formula
We can recognize that the integral \( \int \frac{x^n}{\sqrt{a^2 - x^2}} \, dx \) can be solved using standard integral formulas. In our case, we can use the formula:
\[
\int \frac{x^n}{\sqrt{a^2 - x^2}} \, dx = \frac{x^{n+1}}{n+1} \sqrt{a^2 - x^2} + \frac{a^2}{n+1} \int \frac{x^n}{\sqrt{a^2 - x^2}} \, dx
\]
However, for our specific case, we can directly apply the result for \( n = \frac{1}{3} \).
### Step 6: Solve the Integral
Using the formula, we find:
\[
\int \frac{t^{1/3}}{\sqrt{a^3 - t}} \, dt = \frac{2}{3} \cdot \sin^{-1}\left(\frac{t^{3/2}}{a^{3/2}}\right) + C
\]
### Step 7: Back Substitute
Now we substitute back \( t = x^{3/2} \):
\[
\int \frac{\sqrt{x}}{\sqrt{a^3 - x^3}} \, dx = \frac{2}{3} \sin^{-1}\left(\frac{x^{3/2}}{a^{3/2}}\right) + C
\]
### Final Answer
Thus, the final result of the integral is:
\[
\int \frac{\sqrt{x}}{\sqrt{a^3 - x^3}} \, dx = \frac{2}{3} \sin^{-1}\left(\frac{x^{3/2}}{a^{3/2}}\right) + C
\]
