`int(sqrt(4+x^(2)))/(x^(6))dx=(A(4+x^(2))^(3//2)(Bx^(2)-6))/(x^(5))+C`, then

Options

A. `A=(1)/(120)`
B. `B=1`
C. `A=-(1)/(120)`
D. `B=-1`

Answer

To solve the integral \( \int \frac{\sqrt{4+x^2}}{x^6} \, dx \), we will follow a systematic approach.

### Step 1: Rewrite the Integral
We can rewrite the integral by separating the terms:
\[ \int \frac{\sqrt{4+x^2}}{x^6} \, dx = \int \frac{\sqrt{4+x^2}}{x^2} \cdot \frac{1}{x^4} \, dx \]
This allows us to simplify the expression for easier integration.

### Step 2: Substitute
Let \( t = \sqrt{4+x^2} \). Then, we have:
\[ t^2 = 4 + x^2 \implies x^2 = t^2 - 4 \]
Differentiating both sides gives:
\[ 2t \, dt = 2x \, dx \implies dx = \frac{t}{x} \, dt \]
Now, we need to express \( x \) in terms of \( t \):
\[ x = \sqrt{t^2 - 4} \]
Thus,
\[ dx = \frac{t}{\sqrt{t^2 - 4}} \, dt \]

### Step 3: Substitute into the Integral
Now substituting \( t \) and \( dx \) into the integral:
\[ \int \frac{t}{\sqrt{t^2 - 4}} \cdot \frac{1}{(\sqrt{t^2 - 4})^3} \cdot \frac{t}{\sqrt{t^2 - 4}} \, dt \]
This simplifies to:
\[ \int \frac{t^2}{(t^2 - 4)^{3/2}} \, dt \]

### Step 4: Integrate
Now we can integrate:
\[ \int \frac{t^2}{(t^2 - 4)^{3/2}} \, dt \]
We can use integration by parts or a trigonometric substitution. However, we can also directly integrate:
\[ = -\frac{1}{2(t^2 - 4)^{1/2}} + C \]

### Step 5: Back Substitute
Now substitute back for \( t \):
\[ = -\frac{1}{2\sqrt{4+x^2 - 4}} + C = -\frac{1}{2\sqrt{x^2}} + C = -\frac{1}{2|x|} + C \]

### Step 6: Final Form
Thus, we have:
\[ \int \frac{\sqrt{4+x^2}}{x^6} \, dx = -\frac{1}{2|x|} + C \]

### Comparison
To compare with the given form \( \frac{A(4+x^2)^{3/2}(Bx^2 - 6)}{x^5} + C \), we can identify constants \( A \) and \( B \) based on the coefficients.