∫ 1/√x dx

Steps to Solve

1. Rewrite the integral in exponent form

The integral can be expressed using exponent notation. We know that $ \sqrt{x} = x^{1/2} $, so we rewrite the integrand as:

$$ \frac{1}{\sqrt{x}} = x^{-1/2} $$

This gives us the integral:

$$ \int x^{-1/2} , dx $$

2. Apply the power rule of integration

Using the power rule of integration, which states that:

$$ \int x^n , dx = \frac{x^{n+1}}{n+1} + C $$

for any real number $ n \neq -1 $, we apply it to our integral. Here, $ n = -\frac{1}{2} $:

$$ n + 1 = -\frac{1}{2} + 1 = \frac{1}{2} $$

Now we can evaluate the integral:

$$ \int x^{-1/2} , dx = \frac{x^{1/2}}{1/2} + C = 2x^{1/2} + C $$

3. Write the final result

Thus, the integral of the function $ \frac{1}{\sqrt{x}} $ with respect to $ x $ is:

$$ \int \frac{1}{\sqrt{x}} , dx = 2\sqrt{x} + C $$