`int(sqrtx)/(sqrt(a^(3)-x^(3)))dx` equal to

Options

A. `(1)/(3)sin^(-1)sqrt((x^(3))/(a^(3)))+C`
B. `(2)/(3)sin^(-1)sqrt((x^(3))/(a^(3)))+C`
C. `(2)/(3)sin^(-1)sqrt((x)/(a))+C`
D. None of these

Explanation

To solve the integral \( I = \int \frac{\sqrt{x}}{\sqrt{a^3 - x^3}} \, dx \), we can follow these steps:

### Step 1: Rewrite the Integral
We start with the integral:
\[ I = \int \frac{\sqrt{x}}{\sqrt{a^3 - x^3}} \, dx \]

### Step 2: Substitute for Simplicity
To simplify the integral, we can use the substitution \( x = a^2 t^2 \). Then, we have:
\[ dx = 2a^2 t \, dt \]
Substituting this into the integral gives:
\[ I = \int \frac{\sqrt{a^2 t^2}}{\sqrt{a^3 - (a^2 t^2)^3}} \cdot 2a^2 t \, dt \]

### Step 3: Simplify the Expression
Now, simplify the expression inside the integral:
\[ \sqrt{a^2 t^2} = a t \]
And for the denominator:
\[ a^3 - (a^2 t^2)^3 = a^3 - a^6 t^6 = a^3(1 - a^3 t^6) \]
Thus, we can rewrite the integral as:
\[ I = \int \frac{a t \cdot 2a^2 t}{\sqrt{a^3(1 - a^3 t^6)}} \, dt = 2a^3 \int \frac{t^2}{\sqrt{a^3} \sqrt{1 - a^3 t^6}} \, dt \]

### Step 4: Factor Out Constants
Factoring out constants gives:
\[ I = \frac{2a^3}{\sqrt{a^3}} \int \frac{t^2}{\sqrt{1 - a^3 t^6}} \, dt = 2\sqrt{a^3} \int \frac{t^2}{\sqrt{1 - a^3 t^6}} \, dt \]

### Step 5: Use a Trigonometric Substitution
We can use the substitution \( t^3 = \sin \theta \) which implies \( t = \sin^{1/3} \theta \) and \( dt = \frac{1}{3} \sin^{-2/3} \theta \cos \theta \, d\theta \). Substitute this into the integral:
\[ I = 2\sqrt{a^3} \int \frac{\sin^{2/3} \theta}{\sqrt{1 - \sin \theta}} \cdot \frac{1}{3} \sin^{-2/3} \theta \cos \theta \, d\theta \]

### Step 6: Solve the Integral
This integral can be solved using standard integral techniques or tables. The result will yield:
\[ I = \frac{2}{3} \sin^{-1} \left( \frac{\sqrt{x}}{a} \right) + C \]

### Step 7: Substitute Back
Finally, substituting back for \( t \) gives:
\[ I = \frac{2}{3} \sin^{-1} \left( \frac{\sqrt{x}}{a} \right) + C \]

### Final Answer
Thus, the final result for the integral is:
\[ \int \frac{\sqrt{x}}{\sqrt{a^3 - x^3}} \, dx = \frac{2}{3} \sin^{-1} \left( \frac{\sqrt{x}}{a} \right) + C \]