Evaluate the integral of 1/(√x) dx.

Question image

Understand the Problem

The question is asking to evaluate the integral of the function 1/(√x) with respect to x. This integral is related to basic calculus and will involve finding the antiderivative of the given expression.

Answer

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

The final answer is

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

Steps to Solve

  1. Rewrite the Integral

The given integral is

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

We can rewrite $\frac{1}{\sqrt{x}}$ as $x^{-1/2}$.

  1. Apply the Power Rule for Integration

Using the power rule of integration, which states that

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

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

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

  1. Simplify the Expression

Simplifying the expression from the previous step gives:

$$ \frac{x^{\frac{1}{2}}}{\frac{1}{2}} = 2x^{\frac{1}{2}} $$

So we have:

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

The final answer is

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

More Information

This integral represents the area under the curve of the function $f(x) = \frac{1}{\sqrt{x}}$. The constant $C$ denotes the constant of integration, which is included because the integral represents a family of functions.

Tips

  • Forgetting to add the constant of integration $C$ after evaluating the integral.
  • Misapplying the power rule by incorrectly adding or altering the exponent.

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