Calculus Examples

${(\sqrt{x})}^{2}$

Step 1

Write ${(\sqrt{x})}^{2}$ as a function.

$f (x) = {(\sqrt{x})}^{2}$

Step 2

The function $F (x)$ can be found by finding the indefinite integral of the derivative $f (x)$ .

$F (x) = \int f (x) d x$

Step 3

Set up the integral to solve.

$F (x) = \int {(\sqrt{x})}^{2} d x$

Step 4

Rewrite ${(\sqrt{x})}^{2}$ as $x$ .

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Step 4.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x}$ as $x^{\frac{1}{2}}$ .

$\int {(x^{\frac{1}{2}})}^{2} d x$

Step 4.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\int x^{\frac{1}{2} \cdot 2} d x$

Step 4.3

Combine $\frac{1}{2}$ and $2$ .

$\int x^{\frac{2}{2}} d x$

Step 4.4

Cancel the common factor of $2$ .

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Step 4.4.1

Cancel the common factor.

$\int x^{\frac{2}{2}} d x$

Step 4.4.2

Rewrite the expression.

$\int x^{1} d x$

Step 4.5

Simplify.

$\int x d x$

Step 5

By the Power Rule, the integral of $x$ with respect to $x$ is $\frac{1}{2} x^{2}$ .

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

Step 6

The answer is the antiderivative of the function $f (x) = {(\sqrt{x})}^{2}$ .

$F (x) =$ $\frac{1}{2} x^{2} + C$