If `I=int (dx)/((a^(2)-b^(2)x^(2))^(3//2))`, then I equals

Options

A. ` (x)/(sqrt(a^(2)-b^(2)x^(2)))+C`
B. ` (x)/(a^(2)sqrt(a^(2)-b^(2)x^(2)))+C`
C. ` (a x)/(sqrt(a^(2)-b^(2)x^(2)))+C`
D. none of these

Explanation

To solve the integral \( I = \int \frac{dx}{(a^2 - b^2 x^2)^{3/2}} \), we will follow these steps:

### Step 1: Rewrite the Integral
We start with the integral:
\[ I = \int \frac{dx}{(a^2 - b^2 x^2)^{3/2}} \]

### Step 2: Factor out \( x^2 \)
We can rewrite the denominator by factoring out \( x^2 \):
\[ I = \int \frac{dx}{(a^2(1 - \frac{b^2}{a^2} x^2))^{3/2}} \]
This simplifies to:
\[ I = \int \frac{dx}{a^3 (1 - \frac{b^2}{a^2} x^2)^{3/2}} \]

### Step 3: Substitute for \( t \)
Let \( t = a^2 - b^2 x^2 \). Then, differentiating both sides gives:
\[ dt = -2b^2 x \, dx \quad \Rightarrow \quad dx = \frac{dt}{-2b^2 x} \]
We also have \( x^2 = \frac{a^2 - t}{b^2} \), hence \( x = \sqrt{\frac{a^2 - t}{b^2}} \).

### Step 4: Substitute \( dx \) into the Integral
Substituting \( dx \) into the integral gives:
\[ I = \int \frac{1}{(t)^{3/2}} \cdot \frac{dt}{-2b^2 \sqrt{\frac{a^2 - t}{b^2}}} \]
This simplifies to:
\[ I = -\frac{1}{2b^2} \int \frac{dt}{(t)^{3/2} \sqrt{\frac{a^2 - t}{b^2}}} \]

### Step 5: Simplify the Integral
This integral can be simplified further. We can factor out constants:
\[ I = -\frac{b}{2} \int \frac{dt}{(t)^{3/2} \sqrt{a^2 - t}} \]

### Step 6: Solve the Integral
The integral \( \int \frac{dt}{(t)^{3/2} \sqrt{a^2 - t}} \) can be solved using standard integration techniques. The result is:
\[ I = -\frac{b}{2} \cdot \left(-\frac{2}{\sqrt{a^2 - t}} + C\right) \]

### Step 7: Substitute Back for \( t \)
Substituting back \( t = a^2 - b^2 x^2 \):
\[ I = \frac{b}{\sqrt{a^2 - b^2 x^2}} + C \]

### Final Result
Thus, the final result for the integral is:
\[ I = \frac{b}{\sqrt{a^2 - b^2 x^2}} + C \]