Calculus Examples

$\int \frac{2 - 6 \sqrt{x}}{8 \sqrt[4]{x}} d x$

Step 1

Cancel the common factor of $2 - 6 \sqrt{x}$ and $8$ .

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Step 1.1

Factor $2$ out of $2$ .

$\int \frac{2 (1) - 6 \sqrt{x}}{8 \sqrt[4]{x}} d x$

Step 1.2

Factor $2$ out of $- 6 \sqrt{x}$ .

$\int \frac{2 (1) + 2 (- 3 \sqrt{x})}{8 \sqrt[4]{x}} d x$

Step 1.3

Factor $2$ out of $2 (1) + 2 (- 3 \sqrt{x})$ .

$\int \frac{2 (1 - 3 \sqrt{x})}{8 \sqrt[4]{x}} d x$

Step 1.4

Cancel the common factors.

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Step 1.4.1

Factor $2$ out of $8 \sqrt[4]{x}$ .

$\int \frac{2 (1 - 3 \sqrt{x})}{2 (4 \sqrt[4]{x})} d x$

Step 1.4.2

Cancel the common factor.

$\int \frac{2 (1 - 3 \sqrt{x})}{2 (4 \sqrt[4]{x})} d x$

Step 1.4.3

Rewrite the expression.

$\int \frac{1 - 3 \sqrt{x}}{4 \sqrt[4]{x}} d x$

Step 2

Since $\frac{1}{4}$ is constant with respect to $x$ , move $\frac{1}{4}$ out of the integral.

$\frac{1}{4} \int \frac{1 - 3 \sqrt{x}}{\sqrt[4]{x}} d x$

Step 3

Apply basic rules of exponents.

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Step 3.1

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

$\frac{1}{4} \int \frac{1 - 3 x^{\frac{1}{2}}}{\sqrt[4]{x}} d x$

Step 3.2

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[4]{x}$ as $x^{\frac{1}{4}}$ .

$\frac{1}{4} \int \frac{1 - 3 x^{\frac{1}{2}}}{x^{\frac{1}{4}}} d x$

Step 3.3

Move $x^{\frac{1}{4}}$ out of the denominator by raising it to the $- 1$ power.

$\frac{1}{4} \int (1 - 3 x^{\frac{1}{2}}) {(x^{\frac{1}{4}})}^{- 1} d x$

Step 3.4

Multiply the exponents in ${(x^{\frac{1}{4}})}^{- 1}$ .

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Step 3.4.1

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

$\frac{1}{4} \int (1 - 3 x^{\frac{1}{2}}) x^{\frac{1}{4} \cdot - 1} d x$

Step 3.4.2

Combine $\frac{1}{4}$ and $- 1$ .

$\frac{1}{4} \int (1 - 3 x^{\frac{1}{2}}) x^{\frac{- 1}{4}} d x$

Step 3.4.3

Move the negative in front of the fraction.

$\frac{1}{4} \int (1 - 3 x^{\frac{1}{2}}) x^{- \frac{1}{4}} d x$

Step 4

Simplify.

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Step 4.1

Apply the distributive property.

$\frac{1}{4} \int 1 x^{- \frac{1}{4}} - 3 x^{\frac{1}{2}} x^{- \frac{1}{4}} d x$

Step 4.2

Multiply $x^{- \frac{1}{4}}$ by $1$ .

$\frac{1}{4} \int x^{- \frac{1}{4}} - 3 x^{\frac{1}{2}} x^{- \frac{1}{4}} d x$

Step 4.3

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

$\frac{1}{4} \int x^{- \frac{1}{4}} - 3 x^{\frac{1}{2} - \frac{1}{4}} d x$

Step 4.4

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

$\frac{1}{4} \int x^{- \frac{1}{4}} - 3 x^{\frac{1}{2} \cdot \frac{2}{2} - \frac{1}{4}} d x$

Step 4.5

Write each expression with a common denominator of $4$ , by multiplying each by an appropriate factor of $1$ .

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Step 4.5.1

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

$\frac{1}{4} \int x^{- \frac{1}{4}} - 3 x^{\frac{2}{2 \cdot 2} - \frac{1}{4}} d x$

Step 4.5.2

Multiply $2$ by $2$ .

$\frac{1}{4} \int x^{- \frac{1}{4}} - 3 x^{\frac{2}{4} - \frac{1}{4}} d x$

Step 4.6

Combine the numerators over the common denominator.

$\frac{1}{4} \int x^{- \frac{1}{4}} - 3 x^{\frac{2 - 1}{4}} d x$

Step 4.7

Subtract $1$ from $2$ .

$\frac{1}{4} \int x^{- \frac{1}{4}} - 3 x^{\frac{1}{4}} d x$

Step 5

Split the single integral into multiple integrals.

$\frac{1}{4} (\int x^{- \frac{1}{4}} d x + \int - 3 x^{\frac{1}{4}} d x)$

Step 6

By the Power Rule, the integral of $x^{- \frac{1}{4}}$ with respect to $x$ is $\frac{4}{3} x^{\frac{3}{4}}$ .

$\frac{1}{4} (\frac{4}{3} x^{\frac{3}{4}} + C + \int - 3 x^{\frac{1}{4}} d x)$

Step 7

Since $- 3$ is constant with respect to $x$ , move $- 3$ out of the integral.

$\frac{1}{4} (\frac{4}{3} x^{\frac{3}{4}} + C - 3 \int x^{\frac{1}{4}} d x)$

Step 8

By the Power Rule, the integral of $x^{\frac{1}{4}}$ with respect to $x$ is $\frac{4}{5} x^{\frac{5}{4}}$ .

$\frac{1}{4} (\frac{4}{3} x^{\frac{3}{4}} + C - 3 (\frac{4}{5} x^{\frac{5}{4}} + C))$

Step 9

Simplify.

$\frac{1}{4} (\frac{4 x^{\frac{3}{4}}}{3} - \frac{12 x^{\frac{5}{4}}}{5}) + C$

Step 10

Reorder terms.

$\frac{1}{4} (\frac{4}{3} x^{\frac{3}{4}} - \frac{12}{5} x^{\frac{5}{4}}) + C$