Calculus Examples

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

Step 1

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

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

Step 2

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

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

Step 3

Set up the integral to solve.

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

Step 4

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

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

Step 5

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

$\frac{x}{\sqrt{x}} (\frac{1}{2} d^{2} + C)$

Step 6

Simplify the answer.

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Step 6.1

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

$\frac{x \sqrt{x}}{x} (\frac{1}{2} d^{2}) + C$

Step 6.2

Simplify.

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Step 6.2.1

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

$\frac{x \cdot x^{\frac{1}{2}}}{x} (\frac{1}{2} d^{2}) + C$

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 $1$ .

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

Step 6.2.2.1.2

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

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

Step 6.2.2.2

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

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

Step 6.2.2.3

Combine the numerators over the common denominator.

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

Step 6.2.2.4

Add $2$ and $1$ .

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

Step 6.2.3

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

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

Step 6.2.4

Multiply $x^{\frac{3}{2}}$ by $x^{- 1}$ by adding the exponents.

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Step 6.2.4.1

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

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

Step 6.2.4.2

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$x^{\frac{3}{2} - 1 \cdot \frac{2}{2}} (\frac{1}{2} d^{2}) + C$

Step 6.2.4.3

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

$x^{\frac{3}{2} + \frac{- 1 \cdot 2}{2}} (\frac{1}{2} d^{2}) + C$

Step 6.2.4.4

Combine the numerators over the common denominator.

$x^{\frac{3 - 1 \cdot 2}{2}} (\frac{1}{2} d^{2}) + C$

Step 6.2.4.5

Simplify the numerator.

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Step 6.2.4.5.1

Multiply $- 1$ by $2$ .

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

Step 6.2.4.5.2

Subtract $2$ from $3$ .

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

Step 6.2.5

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

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

Step 6.2.6

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

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

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

Step 7

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

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