Calculus Examples

$\int (\frac{8 - 4 x + 6 x^{\frac{4}{3}} + 12 \sqrt[3]{x^{8}}}{4 \sqrt[3]{x}}) d x$

Step 1

Remove parentheses.

$\int \frac{8 - 4 x + 6 x^{\frac{4}{3}} + 12 \sqrt[3]{x^{8}}}{4 \sqrt[3]{x}} d x$

Step 2

Simplify.

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Step 2.1

Factor $2$ out of $8$ .

$\int \frac{2 (4) - 4 x + 6 x^{\frac{4}{3}} + 12 \sqrt[3]{x^{8}}}{4 \sqrt[3]{x}} d x$

Step 2.2

Factor $2$ out of $- 4 x$ .

$\int \frac{2 (4) + 2 (- 2 x) + 6 x^{\frac{4}{3}} + 12 \sqrt[3]{x^{8}}}{4 \sqrt[3]{x}} d x$

Step 2.3

Factor $2$ out of $2 (4) + 2 (- 2 x)$ .

$\int \frac{2 (4 - 2 x) + 6 x^{\frac{4}{3}} + 12 \sqrt[3]{x^{8}}}{4 \sqrt[3]{x}} d x$

Step 2.4

Factor $2$ out of $6 x^{\frac{4}{3}}$ .

$\int \frac{2 (4 - 2 x) + 2 (3 x^{\frac{4}{3}}) + 12 \sqrt[3]{x^{8}}}{4 \sqrt[3]{x}} d x$

Step 2.5

Factor $2$ out of $2 (4 - 2 x) + 2 (3 x^{\frac{4}{3}})$ .

$\int \frac{2 (4 - 2 x + 3 x^{\frac{4}{3}}) + 12 \sqrt[3]{x^{8}}}{4 \sqrt[3]{x}} d x$

Step 2.6

Factor $2$ out of $12 \sqrt[3]{x^{8}}$ .

$\int \frac{2 (4 - 2 x + 3 x^{\frac{4}{3}}) + 2 (6 \sqrt[3]{x^{8}})}{4 \sqrt[3]{x}} d x$

Step 2.7

Factor $2$ out of $2 (4 - 2 x + 3 x^{\frac{4}{3}}) + 2 (6 \sqrt[3]{x^{8}})$ .

$\int \frac{2 (4 - 2 x + 3 x^{\frac{4}{3}} + 6 \sqrt[3]{x^{8}})}{4 \sqrt[3]{x}} d x$

Step 2.8

Cancel the common factors.

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Step 2.8.1

Factor $2$ out of $4 \sqrt[3]{x}$ .

$\int \frac{2 (4 - 2 x + 3 x^{\frac{4}{3}} + 6 \sqrt[3]{x^{8}})}{2 (2 \sqrt[3]{x})} d x$

Step 2.8.2

Cancel the common factor.

$\int \frac{2 (4 - 2 x + 3 x^{\frac{4}{3}} + 6 \sqrt[3]{x^{8}})}{2 (2 \sqrt[3]{x})} d x$

Step 2.8.3

Rewrite the expression.

$\int \frac{4 - 2 x + 3 x^{\frac{4}{3}} + 6 \sqrt[3]{x^{8}}}{2 \sqrt[3]{x}} d x$

Step 3

Since $\frac{1}{2}$ is constant with respect to $x$ , move $\frac{1}{2}$ out of the integral.

$\frac{1}{2} \int \frac{4 - 2 x + 3 x^{\frac{4}{3}} + 6 \sqrt[3]{x^{8}}}{\sqrt[3]{x}} d x$

Step 4

Apply basic rules of exponents.

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Step 4.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{x^{8}}$ as $x^{\frac{8}{3}}$ .

$\frac{1}{2} \int \frac{4 - 2 x + 3 x^{\frac{4}{3}} + 6 x^{\frac{8}{3}}}{\sqrt[3]{x}} d x$

Step 4.2

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{x}$ as $x^{\frac{1}{3}}$ .

$\frac{1}{2} \int \frac{4 - 2 x + 3 x^{\frac{4}{3}} + 6 x^{\frac{8}{3}}}{x^{\frac{1}{3}}} d x$

Step 4.3

Move $x^{\frac{1}{3}}$ out of the denominator by raising it to the $- 1$ power.

$\frac{1}{2} \int (4 - 2 x + 3 x^{\frac{4}{3}} + 6 x^{\frac{8}{3}}) {(x^{\frac{1}{3}})}^{- 1} d x$

Step 4.4

Multiply the exponents in ${(x^{\frac{1}{3}})}^{- 1}$ .

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Step 4.4.1

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

$\frac{1}{2} \int (4 - 2 x + 3 x^{\frac{4}{3}} + 6 x^{\frac{8}{3}}) x^{\frac{1}{3} \cdot - 1} d x$

Step 4.4.2

Combine $\frac{1}{3}$ and $- 1$ .

$\frac{1}{2} \int (4 - 2 x + 3 x^{\frac{4}{3}} + 6 x^{\frac{8}{3}}) x^{\frac{- 1}{3}} d x$

Step 4.4.3

Move the negative in front of the fraction.

$\frac{1}{2} \int (4 - 2 x + 3 x^{\frac{4}{3}} + 6 x^{\frac{8}{3}}) x^{- \frac{1}{3}} d x$

Step 5

Simplify.

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Step 5.1

Apply the distributive property.

$\frac{1}{2} \int (4 - 2 x + 3 x^{\frac{4}{3}}) x^{- \frac{1}{3}} + 6 x^{\frac{8}{3}} x^{- \frac{1}{3}} d x$

Step 5.2

Apply the distributive property.

$\frac{1}{2} \int (4 - 2 x) x^{- \frac{1}{3}} + 3 x^{\frac{4}{3}} x^{- \frac{1}{3}} + 6 x^{\frac{8}{3}} x^{- \frac{1}{3}} d x$

Step 5.3

Apply the distributive property.

$\frac{1}{2} \int 4 x^{- \frac{1}{3}} - 2 x \cdot x^{- \frac{1}{3}} + 3 x^{\frac{4}{3}} x^{- \frac{1}{3}} + 6 x^{\frac{8}{3}} x^{- \frac{1}{3}} d x$

Step 5.4

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

$\frac{1}{2} \int 4 x^{- \frac{1}{3}} - 2 (x^{1} x^{- \frac{1}{3}}) + 3 x^{\frac{4}{3}} x^{- \frac{1}{3}} + 6 x^{\frac{8}{3}} x^{- \frac{1}{3}} d x$

Step 5.5

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

$\frac{1}{2} \int 4 x^{- \frac{1}{3}} - 2 x^{1 - \frac{1}{3}} + 3 x^{\frac{4}{3}} x^{- \frac{1}{3}} + 6 x^{\frac{8}{3}} x^{- \frac{1}{3}} d x$

Step 5.6

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

$\frac{1}{2} \int 4 x^{- \frac{1}{3}} - 2 x^{\frac{3}{3} - \frac{1}{3}} + 3 x^{\frac{4}{3}} x^{- \frac{1}{3}} + 6 x^{\frac{8}{3}} x^{- \frac{1}{3}} d x$

Step 5.7

Combine the numerators over the common denominator.

$\frac{1}{2} \int 4 x^{- \frac{1}{3}} - 2 x^{\frac{3 - 1}{3}} + 3 x^{\frac{4}{3}} x^{- \frac{1}{3}} + 6 x^{\frac{8}{3}} x^{- \frac{1}{3}} d x$

Step 5.8

Subtract $1$ from $3$ .

$\frac{1}{2} \int 4 x^{- \frac{1}{3}} - 2 x^{\frac{2}{3}} + 3 x^{\frac{4}{3}} x^{- \frac{1}{3}} + 6 x^{\frac{8}{3}} x^{- \frac{1}{3}} d x$

Step 5.9

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

$\frac{1}{2} \int 4 x^{- \frac{1}{3}} - 2 x^{\frac{2}{3}} + 3 x^{\frac{4}{3} - \frac{1}{3}} + 6 x^{\frac{8}{3}} x^{- \frac{1}{3}} d x$

Step 5.10

Combine the numerators over the common denominator.

$\frac{1}{2} \int 4 x^{- \frac{1}{3}} - 2 x^{\frac{2}{3}} + 3 x^{\frac{4 - 1}{3}} + 6 x^{\frac{8}{3}} x^{- \frac{1}{3}} d x$

Step 5.11

Subtract $1$ from $4$ .

$\frac{1}{2} \int 4 x^{- \frac{1}{3}} - 2 x^{\frac{2}{3}} + 3 x^{\frac{3}{3}} + 6 x^{\frac{8}{3}} x^{- \frac{1}{3}} d x$

Step 5.12

Cancel the common factor of $3$ .

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Step 5.12.1

Cancel the common factor.

$\frac{1}{2} \int 4 x^{- \frac{1}{3}} - 2 x^{\frac{2}{3}} + 3 x^{\frac{3}{3}} + 6 x^{\frac{8}{3}} x^{- \frac{1}{3}} d x$

Step 5.12.2

Rewrite the expression.

$\frac{1}{2} \int 4 x^{- \frac{1}{3}} - 2 x^{\frac{2}{3}} + 3 x^{1} + 6 x^{\frac{8}{3}} x^{- \frac{1}{3}} d x$

Step 5.13

Simplify.

$\frac{1}{2} \int 4 x^{- \frac{1}{3}} - 2 x^{\frac{2}{3}} + 3 x + 6 x^{\frac{8}{3}} x^{- \frac{1}{3}} d x$

Step 5.14

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

$\frac{1}{2} \int 4 x^{- \frac{1}{3}} - 2 x^{\frac{2}{3}} + 3 x + 6 x^{\frac{8}{3} - \frac{1}{3}} d x$

Step 5.15

Combine the numerators over the common denominator.

$\frac{1}{2} \int 4 x^{- \frac{1}{3}} - 2 x^{\frac{2}{3}} + 3 x + 6 x^{\frac{8 - 1}{3}} d x$

Step 5.16

Subtract $1$ from $8$ .

$\frac{1}{2} \int 4 x^{- \frac{1}{3}} - 2 x^{\frac{2}{3}} + 3 x + 6 x^{\frac{7}{3}} d x$

Step 5.17

Move $- 2 x^{\frac{2}{3}}$ .

$\frac{1}{2} \int 4 x^{- \frac{1}{3}} + 3 x - 2 x^{\frac{2}{3}} + 6 x^{\frac{7}{3}} d x$

Step 5.18

Reorder $4 x^{- \frac{1}{3}}$ and $3 x$ .

$\frac{1}{2} \int 3 x + 4 x^{- \frac{1}{3}} - 2 x^{\frac{2}{3}} + 6 x^{\frac{7}{3}} d x$

Step 5.19

Move $- 2 x^{\frac{2}{3}}$ .

$\frac{1}{2} \int 3 x + 4 x^{- \frac{1}{3}} + 6 x^{\frac{7}{3}} - 2 x^{\frac{2}{3}} d x$

Step 5.20

Move $4 x^{- \frac{1}{3}}$ .

$\frac{1}{2} \int 3 x + 6 x^{\frac{7}{3}} + 4 x^{- \frac{1}{3}} - 2 x^{\frac{2}{3}} d x$

Step 6

Split the single integral into multiple integrals.

$\frac{1}{2} (\int 3 x d x + \int 6 x^{\frac{7}{3}} d x + \int 4 x^{- \frac{1}{3}} d x + \int - 2 x^{\frac{2}{3}} d x)$

Step 7

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

$\frac{1}{2} (3 \int x d x + \int 6 x^{\frac{7}{3}} d x + \int 4 x^{- \frac{1}{3}} d x + \int - 2 x^{\frac{2}{3}} d x)$

Step 8

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

$\frac{1}{2} (3 (\frac{1}{2} x^{2} + C) + \int 6 x^{\frac{7}{3}} d x + \int 4 x^{- \frac{1}{3}} d x + \int - 2 x^{\frac{2}{3}} d x)$

Step 9

Since $6$ is constant with respect to $x$ , move $6$ out of the integral.

$\frac{1}{2} (3 (\frac{1}{2} x^{2} + C) + 6 \int x^{\frac{7}{3}} d x + \int 4 x^{- \frac{1}{3}} d x + \int - 2 x^{\frac{2}{3}} d x)$

Step 10

By the Power Rule, the integral of $x^{\frac{7}{3}}$ with respect to $x$ is $\frac{3}{10} x^{\frac{10}{3}}$ .

$\frac{1}{2} (3 (\frac{1}{2} x^{2} + C) + 6 (\frac{3}{10} x^{\frac{10}{3}} + C) + \int 4 x^{- \frac{1}{3}} d x + \int - 2 x^{\frac{2}{3}} d x)$

Step 11

Since $4$ is constant with respect to $x$ , move $4$ out of the integral.

$\frac{1}{2} (3 (\frac{1}{2} x^{2} + C) + 6 (\frac{3}{10} x^{\frac{10}{3}} + C) + 4 \int x^{- \frac{1}{3}} d x + \int - 2 x^{\frac{2}{3}} d x)$

Step 12

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

$\frac{1}{2} (3 (\frac{1}{2} x^{2} + C) + 6 (\frac{3}{10} x^{\frac{10}{3}} + C) + 4 (\frac{3}{2} x^{\frac{2}{3}} + C) + \int - 2 x^{\frac{2}{3}} d x)$

Step 13

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

$\frac{1}{2} (3 (\frac{1}{2} x^{2} + C) + 6 (\frac{3}{10} x^{\frac{10}{3}} + C) + 4 (\frac{3}{2} x^{\frac{2}{3}} + C) - 2 \int x^{\frac{2}{3}} d x)$

Step 14

By the Power Rule, the integral of $x^{\frac{2}{3}}$ with respect to $x$ is $\frac{3}{5} x^{\frac{5}{3}}$ .

$\frac{1}{2} (3 (\frac{1}{2} x^{2} + C) + 6 (\frac{3}{10} x^{\frac{10}{3}} + C) + 4 (\frac{3}{2} x^{\frac{2}{3}} + C) - 2 (\frac{3}{5} x^{\frac{5}{3}} + C))$

Step 15

Simplify.

$\frac{1}{2} (\frac{3 x^{2}}{2} + \frac{9 x^{\frac{10}{3}}}{5} + 6 x^{\frac{2}{3}} - \frac{6 x^{\frac{5}{3}}}{5}) + C$

Step 16

Reorder terms.

$\frac{1}{2} (\frac{3}{2} x^{2} + \frac{9}{5} x^{\frac{10}{3}} + 6 x^{\frac{2}{3}} - \frac{6}{5} x^{\frac{5}{3}}) + C$