Question

\(\int_0^a \frac{dx}{\sqrt{x}} \)

• A. \( 2\sqrt{x} \)
• B. \( 2\sqrt{a} \)
• C. \( \sqrt{x} \)
• D. \( \sqrt{a} \)

Hint:

For integrals of the form \( \int x^{n} \, dx \), use the power rule \( \frac{x^{n+1}}{n+1} \).

Answer 1

The Correct Option is B

We are given: \[ I = \int_0^a \frac{dx}{\sqrt{x}} \] This is a standard power rule integral. We can rewrite it as: \[ I = \int_0^a x^{-\frac{1}{2}} \, dx \] The integral of \( x^n \) is \( \frac{x^{n+1}}{n+1} \). Using this formula, we get: \[ I = \left[ 2x^{\frac{1}{2}} \right]_0^a = 2\sqrt{a} - 0 \] Thus, the correct answer is \( 2\sqrt{a} \).