Calculus Examples

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

Step 1

Simplify.

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Step 1.1

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

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

Step 1.2

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

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Step 1.2.1

Move $x^{\frac{1}{2}}$ .

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

Step 1.2.2

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

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Step 1.2.2.1

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

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

Step 1.2.2.2

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

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

Step 1.2.3

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

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

Step 1.2.4

Combine the numerators over the common denominator.

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

Step 1.2.5

Add $1$ and $2$ .

$\int \frac{2 x - x^{\frac{3}{2}}}{\sqrt{x}} d x$

Step 2

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

$\int \frac{2 x - x^{\frac{3}{2}}}{x^{\frac{1}{2}}} d x$

Step 3

Simplify.

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Step 3.1

Factor $x$ out of $2 x - x^{\frac{3}{2}}$ .

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Step 3.1.1

Factor $x$ out of $2 x$ .

$\int \frac{x \cdot 2 - x^{\frac{3}{2}}}{x^{\frac{1}{2}}} d x$

Step 3.1.2

Factor $x$ out of $- x^{\frac{3}{2}}$ .

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

Step 3.1.3

Factor $x$ out of $x \cdot 2 + x (- x^{\frac{1}{2}})$ .

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

Step 3.2

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

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

Step 3.3

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

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Step 3.3.1

Move $x^{- \frac{1}{2}}$ .

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

Step 3.3.2

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

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Step 3.3.2.1

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

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

Step 3.3.2.2

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

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

Step 3.3.3

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

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

Step 3.3.4

Combine the numerators over the common denominator.

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

Step 3.3.5

Add $- 1$ and $2$ .

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

Step 4

Simplify.

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Step 4.1

Apply the distributive property.

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

Step 4.2

Reorder $x^{\frac{1}{2}}$ and $2$ .

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

Step 4.3

Reorder $x^{\frac{1}{2}}$ and $- 1$ .

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

Step 4.4

Factor out negative.

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

Step 4.5

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

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

Step 4.6

Combine the numerators over the common denominator.

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

Step 4.7

Add $1$ and $1$ .

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

Step 4.8

Cancel the common factor of $2$ .

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Step 4.8.1

Cancel the common factor.

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

Step 4.8.2

Rewrite the expression.

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

Step 4.9

Simplify.

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

Step 4.10

Reorder $2 x^{\frac{1}{2}}$ and $- x$ .

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

Step 5

Split the single integral into multiple integrals.

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

Step 6

Since $- 1$ is constant with respect to $x$ , move $- 1$ out of the integral.

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

Step 7

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

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

Step 8

Since $2$ is constant with respect to $x$ , move $2$ out of the integral.

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

Step 9

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

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

Step 10

Simplify.

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Step 10.1

Simplify.

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

Step 10.2

Simplify.

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Step 10.2.1

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

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

Step 10.2.2

Multiply $2$ by $2$ .

$- \frac{1}{2} x^{2} + \frac{4}{3} x^{\frac{3}{2}} + C$