Calculus Examples

$\frac{1}{2 \sqrt{x}} d x$

Step 1

Write $\frac{1}{2 \sqrt{x}} d x$ as a function.

$f (x) = \frac{1}{2 \sqrt{x}} d x$

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.

$F (x) = \int \frac{1}{2 \sqrt{x}} \cdot (d x) d x$

Step 4

Simplify.

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Step 4.1

Combine $d$ and $\frac{1}{2 \sqrt{x}}$ .

$\int \frac{d}{2 \sqrt{x}} \cdot x d x$

Step 4.2

Combine $\frac{d}{2 \sqrt{x}}$ and $x$ .

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

Step 5

Since $\frac{d}{2}$ is constant with respect to $x$ , move $\frac{d}{2}$ out of the integral.

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

Step 6

Simplify the expression.

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Step 6.1

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

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

Step 6.2

Simplify.

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Step 6.2.1

Move $x^{\frac{1}{2}}$ to the numerator using the negative exponent rule $\frac{1}{b^{n}} = b^{- n}$ .

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

Step 6.2.2

Multiply $x$ by $x^{- \frac{1}{2}}$ by adding the exponents.

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Step 6.2.2.1

Multiply $x$ by $x^{- \frac{1}{2}}$ .

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Step 6.2.2.1.1

Raise $x$ to the power of $1$ .

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

Step 6.2.2.1.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Step 6.2.2.2

Write $1$ as a fraction with a common denominator.

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

Step 6.2.2.3

Combine the numerators over the common denominator.

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

Step 6.2.2.4

Subtract $1$ from $2$ .

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

Step 7

By the Power Rule, the integral of $x^{\frac{1}{2}}$ with respect to $x$ is $\frac{2}{3} x^{\frac{3}{2}}$ .

$\frac{d}{2} (\frac{2}{3} x^{\frac{3}{2}} + C)$

Step 8

Simplify the answer.

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Step 8.1

Combine $\frac{2}{3}$ and $x^{\frac{3}{2}}$ .

$\frac{d}{2} (\frac{2 x^{\frac{3}{2}}}{3} + C)$

Step 8.2

Rewrite $\frac{d}{2} (\frac{2 x^{\frac{3}{2}}}{3} + C)$ as $\frac{1}{2} d \frac{2}{3} x^{\frac{3}{2}} + C$ .

$\frac{1}{2} d \frac{2}{3} x^{\frac{3}{2}} + C$

Step 8.3

Simplify.

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Step 8.3.1

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

$\frac{d}{2} \cdot \frac{2}{3} x^{\frac{3}{2}} + C$

Step 8.3.2

Multiply $\frac{d}{2}$ by $\frac{2}{3}$ .

$\frac{d \cdot 2}{2 \cdot 3} x^{\frac{3}{2}} + C$

Step 8.3.3

Multiply $2$ by $3$ .

$\frac{d \cdot 2}{6} x^{\frac{3}{2}} + C$

Step 8.3.4

Move $2$ to the left of $d$ .

$\frac{2 \cdot d}{6} x^{\frac{3}{2}} + C$

Step 8.3.5

Cancel the common factor of $2$ and $6$ .

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Step 8.3.5.1

Factor $2$ out of $2 d$ .

$\frac{2 (d)}{6} x^{\frac{3}{2}} + C$

Step 8.3.5.2

Cancel the common factors.

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Step 8.3.5.2.1

Factor $2$ out of $6$ .

$\frac{2 d}{2 \cdot 3} x^{\frac{3}{2}} + C$

Step 8.3.5.2.2

Cancel the common factor.

$\frac{2 d}{2 \cdot 3} x^{\frac{3}{2}} + C$

Step 8.3.5.2.3

Rewrite the expression.

$\frac{d}{3} x^{\frac{3}{2}} + C$

Step 8.3.6

Combine $\frac{d}{3}$ and $x^{\frac{3}{2}}$ .

$\frac{d x^{\frac{3}{2}}}{3} + C$

$\frac{1}{3} d x^{\frac{3}{2}} + C$

Step 9

The answer is the antiderivative of the function $f (x) = \frac{1}{2 \sqrt{x}} \cdot (d x)$ .

$F (x) =$ $\frac{1}{3} d x^{\frac{3}{2}} + C$