∫ 1/√x dx

Question image

Understand the Problem

The question is asking for the evaluation of the integral of the function 1 divided by the square root of x with respect to x. This involves using integration techniques to find the antiderivative of the function.

Answer

The integral evaluates to $2\sqrt{x} + C$.
Answer for screen readers

The result of the integral is

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

Steps to Solve

  1. Rewrite the integral

The integral we want to evaluate is

$$ \int \frac{1}{\sqrt{x}} , dx $$

We can rewrite $\frac{1}{\sqrt{x}}$ as $x^{-1/2}$ to simplify the integration process.

  1. Apply the power rule for integration

Using the power rule for integration, which states that

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

for $n \neq -1$, we set $n = -\frac{1}{2}$.

  1. Calculate the antiderivative

Following the power rule, we get:

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

This simplifies to:

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

  1. Final result

Thus, the antiderivative of the original integral is:

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

where $C$ is the constant of integration.

The result of the integral is

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

More Information

This integral is a common example in calculus for introducing the power rule. The function $\frac{1}{\sqrt{x}}$ is particularly important in applications related to area under curves and rates of change.

Tips

  • Forgetting to use the power rule and attempting to evaluate the integral without rewriting the integrand.
  • Neglecting to add the constant of integration $C$ after integration.

AI-generated content may contain errors. Please verify critical information

Thank you for voting!
Use Quizgecko on...
Browser
Browser