Understanding the Integral of √(1/x)
To solve the integral of √(1/x), we start by rewriting the expression in a more manageable form.

Step 1: Rewrite the Integral
- The expression √(1/x) can be rewritten as:
- √(1/x) = (1/x)^(1/2) = x^(-1/2)
Thus, we need to evaluate the integral:
- ∫√(1/x) dx = ∫x^(-1/2) dx

Step 2: Apply the Power Rule of Integration
- The power rule states that ∫x^n dx = (x^(n+1))/(n+1) + C, where n ≠ -1.
For our integral:
- n = -1/2
- n + 1 = -1/2 + 1 = 1/2
Now, we apply the power rule:
- ∫x^(-1/2) dx = (x^(1/2))/(1/2) + C
- = 2x^(1/2) + C

Step 3: Final Result
Thus, the integral of √(1/x) is:
- ∫√(1/x) dx = 2√x + C
where C is the constant of integration.

Conclusion
- To summarize, the integral of √(1/x) leads to a straightforward application of the power rule, resulting in the final expression 2√x + C. This method is essential for solving similar integrals encountered in calculus, especially in UPSC-level examinations.