Question

$\int \frac{dx}{\sqrt{x}} =$

• A. $-2 \sqrt{x} + c$
• B. $\sqrt{x} + c$
• C. $2 \sqrt{x} + c$
• D. $x + c$

Hint:

It is helpful to memorize that the derivative of $\sqrt{x}$ is $\frac{1}{2\sqrt{x}}$. Therefore, by the reverse process of integration, the integral of $\frac{1}{\sqrt{x}}$ must be $2\sqrt{x}$.

Answer 1

The Correct Option is C

1. Rewrite the Expression: The integrand can be written in exponential form: $$\frac{1}{\sqrt{x}} = \frac{1}{x^{1/2}} = x^{-1/2}$$

2. Apply the Power Rule: The rule states $\int x^n \, dx = \frac{x^{n+1}}{n+1} + c$ for $n \neq -1$. Here, $n = -1/2$: $$\int x^{-1/2} \, dx = \frac{x^{-1/2 + 1}}{-1/2 + 1} + c\lt strong\gt 3. Simplify the Fractions:\lt /strong\gt \int x^{-1/2} \, dx = \frac{x^{1/2}}{1/2} + c$$ $$\text{Integral} = 2x^{1/2} + c$$

4. Final Form: Converting the exponent back to radical form: $$\text{Integral} = 2\sqrt{x} + c$$