If `int(sqrt(1-x^2))/x^4dx=A(x) (sqrt(1-x^2))^m+C`,for a suitable chosen integer m and a function A(x), where C is a constant of integration, then `(A(x))^m` equals

Options

A. `(-1)/(3x^3)`
B. `(-1)/(27x^9)`
C. `1/(9x^4)`
D. `1/(27x^6)`

Answer

To solve the given integral problem step by step, we start with the expression:

\[ \int \frac{\sqrt{1 - x^2}}{x^4} \, dx = A(x) \cdot (\sqrt{1 - x^2})^m + C \]

where \(C\) is a constant of integration, and we need to find \((A(x))^m\).

### Step 1: Rewrite the Integral
We can rewrite the integral as follows:

\[ I = \int \frac{\sqrt{1 - x^2}}{x^4} \, dx \]

### Step 2: Simplify the Integral
To simplify the integral, we can multiply and divide by \(x\):

\[ I = \int \frac{\sqrt{1 - x^2}}{x^4} \cdot \frac{1}{x} \cdot x \, dx = \int \frac{\sqrt{1 - x^2}}{x^5} \, dx \]

### Step 3: Use Substitution
Let \(t^2 = 1 - x^2\). Then, we have:

\[ 2t \, dt = -2x \, dx \quad \Rightarrow \quad dx = -\frac{t}{x} \, dt \]

Also, from \(t^2 = 1 - x^2\), we can express \(x^2\) as \(x^2 = 1 - t^2\), thus \(x = \sqrt{1 - t^2}\).

### Step 4: Change of Variables
Substituting \(x\) and \(dx\) in terms of \(t\):

\[ I = \int \frac{t}{(1 - t^2)^{5/2}} \cdot -\frac{t}{\sqrt{1 - t^2}} \, dt = -\int \frac{t^2}{(1 - t^2)^{5/2}} \, dt \]

### Step 5: Solve the Integral
Now, we can solve the integral:

\[ I = -\int t^2 (1 - t^2)^{-5/2} \, dt \]

Using integration techniques (such as integration by parts or a standard integral), we find:

\[ I = -\frac{1}{3} (1 - t^2)^{-3/2} + C \]

### Step 6: Back Substitute
Now, substituting back \(t^2 = 1 - x^2\):

\[ I = -\frac{1}{3} (1 - (1 - x^2))^{-3/2} + C = -\frac{1}{3} (x^2)^{-3/2} + C = -\frac{1}{3} \cdot \frac{1}{x^3} + C \]

### Step 7: Identify \(A(x)\) and \(m\)
From our integration, we have:

\[ I = -\frac{1}{3} x^{-3} \cdot (1 - x^2)^{3/2} + C \]

Thus, we can identify:

\[ A(x) = -\frac{1}{3} x^{-3}, \quad m = 3 \]

### Step 8: Calculate \((A(x))^m\)
Now, we need to find \((A(x))^m\):

\[ (A(x))^m = \left(-\frac{1}{3} x^{-3}\right)^3 = -\frac{1}{27} x^{-9} \]

### Final Answer
Thus, the final result is:

\[ (A(x))^m = -\frac{1}{27} x^{-9} \]