∫ 1/√x dx

Question image

Understand the Problem

The question is asking us to solve the integral of the function 1/√x with respect to x. This involves applying integral calculus techniques to find the antiderivative of the given function.

Answer

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

The integral of ( \frac{1}{\sqrt{x}} ) with respect to ( x ) is:

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

Steps to Solve

  1. Rewrite the integral in exponent form

The function ( \frac{1}{\sqrt{x}} ) can be rewritten as ( x^{-\frac{1}{2}} ).

Thus, the integral becomes:

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

  1. Apply the power rule for integration

The power rule states that:

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

where ( n \neq -1 ). Here, ( n = -\frac{1}{2} ), so:

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

  1. Integrate using the power rule

Now apply the power rule:

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

Which simplifies to:

$$ 2x^{\frac{1}{2}} + C $$

  1. Rewrite in terms of the square root

Since ( x^{\frac{1}{2}} ) is the same as ( \sqrt{x} ), we can write:

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

The integral of ( \frac{1}{\sqrt{x}} ) with respect to ( x ) is:

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

More Information

This integral is a fundamental result in calculus, often encountered when working with power functions. The constant ( C ) represents the constant of integration, which appears because indefinite integrals can have infinitely many solutions differing by a constant.

Tips

  • Forgetting to add the constant ( C ) after integrating.
  • Misapplying the power rule when calculating the new exponent and denominator.
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