Question

\( \int \left(\sqrt{x} + \frac{1}{\sqrt{x}}\right)^2 dx = \)

• A. \( \frac{x^2}{2} + 2x + \log|x| + C \)
• B. \( \frac{x^2}{2} + 2 + \log|x| + C \)
• C. \( \frac{x^2}{2} + x + \log|x| + C \)
• D. \( \frac{x^2}{2} + 2x + 2\log|x| + C \)
• E. \( \frac{x^2}{2} - 2x + \log|x| + C \)

Hint:

Always expand before integrating when squares are involved.

Answer 1

The Correct Option is A

Concept:
• First expand the square.
• Then integrate term-by-term.

Step 1: Rewrite expression in powers.
\[ \sqrt{x} = x^{1/2}, \quad \frac{1}{\sqrt{x}} = x^{-1/2} \] So: \[ \left(\sqrt{x} + \frac{1}{\sqrt{x}}\right)^2 = (x^{1/2} + x^{-1/2})^2 \]

Step 2: Expand using identity.
\[ (a+b)^2 = a^2 + 2ab + b^2 \] \[ = x + 2( x^{1/2} \cdot x^{-1/2} ) + \frac{1}{x} \] \[ = x + 2(1) + \frac{1}{x} \] \[ = x + 2 + \frac{1}{x} \]

Step 3: Integrate term-by-term.
\[ \int \left(x + 2 + \frac{1}{x}\right) dx \] \[ = \int x\,dx + \int 2\,dx + \int \frac{1}{x} dx \]

Step 4: Compute each integral.
\[ \int x\,dx = \frac{x^2}{2} \] \[ \int 2\,dx = 2x \] \[ \int \frac{1}{x} dx = \log|x| \]

Step 5: Combine results.
\[ = \frac{x^2}{2} + 2x + \log|x| + C \]

Step 6: Final Answer.
\[ \boxed{\frac{x^2}{2} + 2x + \log|x| + C} \]