∫ 1/√x dx

Question image

Understand the Problem

The question is asking for the integral of the function 1 over the square root of x with respect to x, which requires performing integration techniques.

Answer

$$ 2\sqrt{x} + C $$
Answer for screen readers

$$ 2\sqrt{x} + C $$

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:

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

  1. 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:

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

  1. Write the final result

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

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

$$ 2\sqrt{x} + C $$

More Information

The result of this integral shows that the area under the curve of the function $ \frac{1}{\sqrt{x}} $ with respect to $ x $ grows proportionally to $ \sqrt{x} $. This has applications in various fields such as physics, particularly in problems involving rates of change.

Tips

  • Forgetting to add the constant of integration, $ C $.
  • Misapplying the power rule, especially with negative exponents.
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