The integral `int(dx)/((a^(2)-b^(2)x^(2))^(3//2))` 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. `(ax)/(sqrt(a^(2)-b^(2)x^(2)))+C`
D. `(1)/(a^(2)sqrt(a^(2)-b^(2)x^(2)))+C`

Answer

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: Substitute
Let \( t = a^2 - b^2 x^2 \). Then, differentiating both sides gives:

\[ dt = -2b^2 x \, dx \quad \Rightarrow \quad dx = \frac{dt}{-2b^2 x}. \]

From our substitution, we can express \( x \) in terms of \( t \):

\[ x^2 = \frac{a^2 - t}{b^2} \quad \Rightarrow \quad x = \sqrt{\frac{a^2 - t}{b^2}} = \frac{\sqrt{a^2 - t}}{b}. \]

### Step 3: Substitute \( dx \) in the Integral
Now we substitute \( dx \) and \( x \) back into the integral:

\[ I = \int \frac{1}{t^{3/2}} \cdot \frac{dt}{-2b^2 \cdot \frac{\sqrt{a^2 - t}}{b}}. \]

This simplifies to:

\[ I = \int \frac{-b}{2b^2} \cdot \frac{dt}{t^{3/2} \sqrt{a^2 - t}} = \frac{-1}{2b} \int \frac{dt}{t^{3/2} \sqrt{a^2 - t}}. \]

### Step 4: Solve the Integral
The integral

\[ \int \frac{dt}{t^{3/2} \sqrt{a^2 - t}} \]

can be solved using a standard integral formula. The result is:

\[ \int \frac{dt}{t^{3/2} \sqrt{a^2 - t}} = -\frac{2}{\sqrt{a^2 - t}}. \]

### Step 5: Substitute Back
Now substituting back into our expression for \( I \):

\[ I = \frac{-1}{2b} \left(-\frac{2}{\sqrt{a^2 - t}}\right) + C. \]

This simplifies to:

\[ I = \frac{1}{b \sqrt{a^2 - t}} + C. \]

Substituting \( t = a^2 - b^2 x^2 \) back in gives:

\[ I = \frac{1}{b \sqrt{a^2 - (a^2 - b^2 x^2)}} + C = \frac{1}{b \sqrt{b^2 x^2}} + C = \frac{1}{b |bx|} + C. \]

### Final Result
Thus, the integral evaluates to:

\[ I = \frac{x}{a^2 \sqrt{a^2 - b^2 x^2}} + C. \]