Question

Evaluate the integral: \( \int \frac{1}{x^3} \sqrt{1 - \frac{1}{x^2}} \text{dx} = \)

• A. \( \frac{-1}{6}\left(1 - \frac{1}{x^2}\right)^{\frac{3}{2}} + \text{C} \)
• B. \( \frac{1}{3}\left(1 - \frac{1}{x^2}\right)^{\frac{3}{2}} + \text{C} \)
• C. \( \frac{-1}{3}\left(1 - \frac{1}{x^2}\right)^{\frac{3}{2}} + \text{C} \)
• D. \( \frac{4}{3}\left(1 - \frac{1}{x^2}\right)^{\frac{3}{2}} + \text{C} \)
• E. \( \frac{-4}{3}\left(1 - \frac{1}{x^2}\right)^{\frac{3}{2}} + \text{C} \)

Hint:

Always look for a function and its derivative when dealing with complex integrals. Factoring out terms or rewriting fractions with negative exponents (like $1/x^2$ as $x^{-2}$) often reveals a clear substitution path.

Answer 1

The Correct Option is B

Step 1: Understanding the Concept:
We are given an indefinite integral that involves a composite function under a square root and an algebraic fraction.
The presence of \( 1/x^2 \) inside the root and \( 1/x^3 \) outside strongly suggests the method of substitution, as the derivative of \( x^{-2} \) yields a term proportional to \( x^{-3} \).

Step 2: Key Formula or Approach:
Use the integration by substitution method: Let \( u = g(x) \), then \( du = g'(x)dx \).

Step 3: Detailed Explanation:
Let \( u = 1 - \frac{1}{x^2} = 1 - x^{-2} \).
Differentiating both sides with respect to \( x \), we get:
\[ \frac{du}{dx} = 0 - (-2)x^{-3} = \frac{2}{x^3} \] This implies that \( du = \frac{2}{x^3} dx \).
We can rearrange this to match the term in our integral:
\[ \frac{1}{x^3} dx = \frac{1}{2} du \] Now, substitute \( u \) and \( du \) back into the original integral:
\[ I = \int \sqrt{1 - \frac{1}{x^2}} \cdot \left( \frac{1}{x^3} dx \right) \] \[ I = \int \sqrt{u} \cdot \left( \frac{1}{2} du \right) \] \[ I = \frac{1}{2} \int u^{\frac{1}{2}} du \] Integrate using the power rule \( \int u^n du = \frac{u^{n+1}}{n+1} \):
\[ I = \frac{1}{2} \left[ \frac{u^{\frac{1}{2} + 1}}{\frac{1}{2} + 1} \right] + C \] \[ I = \frac{1}{2} \left[ \frac{u^{\frac{3}{2}}}{\frac{3}{2}} \right] + C \] \[ I = \frac{1}{2} \cdot \frac{2}{3} u^{\frac{3}{2}} + C \] \[ I = \frac{1}{3} u^{\frac{3}{2}} + C \] Finally, substitute back the original expression for \( u \):
\[ I = \frac{1}{3} \left( 1 - \frac{1}{x^2} \right)^{\frac{3}{2}} + C \]

Step 4: Final Answer:
The calculated integral matches option (B).