Calculus Examples

$\int_{0}^{8} 8 - 4 \sqrt[3]{x} d x$

Step 1

Split the single integral into multiple integrals.

$\int_{0}^{8} 8 d x + \int_{0}^{8} - 4 \sqrt[3]{x} d x$

Step 2

Apply the constant rule.

$8 x]_{0}^{8} + \int_{0}^{8} - 4 \sqrt[3]{x} d x$

Step 3

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

$8 x]_{0}^{8} - 4 \int_{0}^{8} \sqrt[3]{x} d x$

Step 4

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

$8 x]_{0}^{8} - 4 \int_{0}^{8} x^{\frac{1}{3}} d x$

Step 5

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

$8 x]_{0}^{8} - 4 (\frac{3}{4} x^{\frac{4}{3}}]_{0}^{8})$

Step 6

Simplify the answer.

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Step 6.1

Combine $\frac{3}{4}$ and $x^{\frac{4}{3}}$ .

$8 x]_{0}^{8} - 4 (\frac{3 x^{\frac{4}{3}}}{4}]_{0}^{8})$

Step 6.2

Substitute and simplify.

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Step 6.2.1

Evaluate $8 x$ at $8$ and at $0$ .

$(8 \cdot 8) - 8 \cdot 0 - 4 (\frac{3 x^{\frac{4}{3}}}{4}]_{0}^{8})$

Step 6.2.2

Evaluate $\frac{3 x^{\frac{4}{3}}}{4}$ at $8$ and at $0$ .

$8 \cdot 8 - 8 \cdot 0 - 4 (\frac{3 \cdot 8^{\frac{4}{3}}}{4} - \frac{3 \cdot 0^{\frac{4}{3}}}{4})$

Step 6.2.3

Simplify.

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Step 6.2.3.1

Multiply $8$ by $8$ .

$64 - 8 \cdot 0 - 4 (\frac{3 \cdot 8^{\frac{4}{3}}}{4} - \frac{3 \cdot 0^{\frac{4}{3}}}{4})$

Step 6.2.3.2

Multiply $- 8$ by $0$ .

$64 + 0 - 4 (\frac{3 \cdot 8^{\frac{4}{3}}}{4} - \frac{3 \cdot 0^{\frac{4}{3}}}{4})$

Step 6.2.3.3

Add $64$ and $0$ .

$64 - 4 (\frac{3 \cdot 8^{\frac{4}{3}}}{4} - \frac{3 \cdot 0^{\frac{4}{3}}}{4})$

Step 6.2.3.4

Rewrite $8$ as $2^{3}$ .

$64 - 4 (\frac{3 \cdot {(2^{3})}^{\frac{4}{3}}}{4} - \frac{3 \cdot 0^{\frac{4}{3}}}{4})$

Step 6.2.3.5

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

$64 - 4 (\frac{3 \cdot 2^{3 (\frac{4}{3})}}{4} - \frac{3 \cdot 0^{\frac{4}{3}}}{4})$

Step 6.2.3.6

Cancel the common factor of $3$ .

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Step 6.2.3.6.1

Cancel the common factor.

$64 - 4 (\frac{3 \cdot 2^{3 (\frac{4}{3})}}{4} - \frac{3 \cdot 0^{\frac{4}{3}}}{4})$

Step 6.2.3.6.2

Rewrite the expression.

$64 - 4 (\frac{3 \cdot 2^{4}}{4} - \frac{3 \cdot 0^{\frac{4}{3}}}{4})$

Step 6.2.3.7

Raise $2$ to the power of $4$ .

$64 - 4 (\frac{3 \cdot 16}{4} - \frac{3 \cdot 0^{\frac{4}{3}}}{4})$

Step 6.2.3.8

Multiply $3$ by $16$ .

$64 - 4 (\frac{48}{4} - \frac{3 \cdot 0^{\frac{4}{3}}}{4})$

Step 6.2.3.9

Cancel the common factor of $48$ and $4$ .

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Step 6.2.3.9.1

Factor $4$ out of $48$ .

$64 - 4 (\frac{4 \cdot 12}{4} - \frac{3 \cdot 0^{\frac{4}{3}}}{4})$

Step 6.2.3.9.2

Cancel the common factors.

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Step 6.2.3.9.2.1

Factor $4$ out of $4$ .

$64 - 4 (\frac{4 \cdot 12}{4 (1)} - \frac{3 \cdot 0^{\frac{4}{3}}}{4})$

Step 6.2.3.9.2.2

Cancel the common factor.

$64 - 4 (\frac{4 \cdot 12}{4 \cdot 1} - \frac{3 \cdot 0^{\frac{4}{3}}}{4})$

Step 6.2.3.9.2.3

Rewrite the expression.

$64 - 4 (\frac{12}{1} - \frac{3 \cdot 0^{\frac{4}{3}}}{4})$

Step 6.2.3.9.2.4

Divide $12$ by $1$ .

$64 - 4 (12 - \frac{3 \cdot 0^{\frac{4}{3}}}{4})$

Step 6.2.3.10

Rewrite $0$ as $0^{3}$ .

$64 - 4 (12 - \frac{3 \cdot {(0^{3})}^{\frac{4}{3}}}{4})$

Step 6.2.3.11

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

$64 - 4 (12 - \frac{3 \cdot 0^{3 (\frac{4}{3})}}{4})$

Step 6.2.3.12

Cancel the common factor of $3$ .

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Step 6.2.3.12.1

Cancel the common factor.

$64 - 4 (12 - \frac{3 \cdot 0^{3 (\frac{4}{3})}}{4})$

Step 6.2.3.12.2

Rewrite the expression.

$64 - 4 (12 - \frac{3 \cdot 0^{4}}{4})$

Step 6.2.3.13

Raising $0$ to any positive power yields $0$ .

$64 - 4 (12 - \frac{3 \cdot 0}{4})$

Step 6.2.3.14

Multiply $3$ by $0$ .

$64 - 4 (12 - \frac{0}{4})$

Step 6.2.3.15

Cancel the common factor of $0$ and $4$ .

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Step 6.2.3.15.1

Factor $4$ out of $0$ .

$64 - 4 (12 - \frac{4 (0)}{4})$

Step 6.2.3.15.2

Cancel the common factors.

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Step 6.2.3.15.2.1

Factor $4$ out of $4$ .

$64 - 4 (12 - \frac{4 \cdot 0}{4 \cdot 1})$

Step 6.2.3.15.2.2

Cancel the common factor.

$64 - 4 (12 - \frac{4 \cdot 0}{4 \cdot 1})$

Step 6.2.3.15.2.3

Rewrite the expression.

$64 - 4 (12 - \frac{0}{1})$

Step 6.2.3.15.2.4

Divide $0$ by $1$ .

$64 - 4 (12 - 0)$

Step 6.2.3.16

Multiply $- 1$ by $0$ .

$64 - 4 (12 + 0)$

Step 6.2.3.17

Add $12$ and $0$ .

$64 - 4 \cdot 12$

Step 6.2.3.18

Multiply $- 4$ by $12$ .

$64 - 48$

Step 6.2.3.19

Subtract $48$ from $64$ .

$16$

Step 7