Calculus Examples

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

Step 1

Split the single integral into multiple integrals.

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

Step 2

Apply the constant rule.

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

Step 3

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

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

Step 4

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

$8 x]_{0}^{8} + \frac{3}{4} x^{\frac{4}{3}}]_{0}^{8}$

Step 5

Simplify the answer.

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Step 5.1

Combine $8 x]_{0}^{8}$ and $\frac{3}{4} x^{\frac{4}{3}}]_{0}^{8}$ .

$8 x + \frac{3}{4} x^{\frac{4}{3}}]_{0}^{8}$

Step 5.2

Substitute and simplify.

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Step 5.2.1

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

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

Step 5.2.2

Simplify.

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Step 5.2.2.1

Multiply $8$ by $8$ .

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

Step 5.2.2.2

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

$64 + \frac{3}{4} \cdot {(2^{3})}^{\frac{4}{3}} - (8 \cdot 0 + \frac{3}{4} \cdot 0^{\frac{4}{3}})$

Step 5.2.2.3

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

$64 + \frac{3}{4} \cdot 2^{3 (\frac{4}{3})} - (8 \cdot 0 + \frac{3}{4} \cdot 0^{\frac{4}{3}})$

Step 5.2.2.4

Cancel the common factor of $3$ .

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Step 5.2.2.4.1

Cancel the common factor.

$64 + \frac{3}{4} \cdot 2^{3 (\frac{4}{3})} - (8 \cdot 0 + \frac{3}{4} \cdot 0^{\frac{4}{3}})$

Step 5.2.2.4.2

Rewrite the expression.

$64 + \frac{3}{4} \cdot 2^{4} - (8 \cdot 0 + \frac{3}{4} \cdot 0^{\frac{4}{3}})$

Step 5.2.2.5

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

$64 + \frac{3}{4} \cdot 16 - (8 \cdot 0 + \frac{3}{4} \cdot 0^{\frac{4}{3}})$

Step 5.2.2.6

Combine $\frac{3}{4}$ and $16$ .

$64 + \frac{3 \cdot 16}{4} - (8 \cdot 0 + \frac{3}{4} \cdot 0^{\frac{4}{3}})$

Step 5.2.2.7

Multiply $3$ by $16$ .

$64 + \frac{48}{4} - (8 \cdot 0 + \frac{3}{4} \cdot 0^{\frac{4}{3}})$

Step 5.2.2.8

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

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Step 5.2.2.8.1

Factor $4$ out of $48$ .

$64 + \frac{4 \cdot 12}{4} - (8 \cdot 0 + \frac{3}{4} \cdot 0^{\frac{4}{3}})$

Step 5.2.2.8.2

Cancel the common factors.

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Step 5.2.2.8.2.1

Factor $4$ out of $4$ .

$64 + \frac{4 \cdot 12}{4 (1)} - (8 \cdot 0 + \frac{3}{4} \cdot 0^{\frac{4}{3}})$

Step 5.2.2.8.2.2

Cancel the common factor.

$64 + \frac{4 \cdot 12}{4 \cdot 1} - (8 \cdot 0 + \frac{3}{4} \cdot 0^{\frac{4}{3}})$

Step 5.2.2.8.2.3

Rewrite the expression.

$64 + \frac{12}{1} - (8 \cdot 0 + \frac{3}{4} \cdot 0^{\frac{4}{3}})$

Step 5.2.2.8.2.4

Divide $12$ by $1$ .

$64 + 12 - (8 \cdot 0 + \frac{3}{4} \cdot 0^{\frac{4}{3}})$

Step 5.2.2.9

Add $64$ and $12$ .

$76 - (8 \cdot 0 + \frac{3}{4} \cdot 0^{\frac{4}{3}})$

Step 5.2.2.10

Multiply $8$ by $0$ .

$76 - (0 + \frac{3}{4} \cdot 0^{\frac{4}{3}})$

Step 5.2.2.11

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

$76 - (0 + \frac{3}{4} \cdot {(0^{3})}^{\frac{4}{3}})$

Step 5.2.2.12

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

$76 - (0 + \frac{3}{4} \cdot 0^{3 (\frac{4}{3})})$

Step 5.2.2.13

Cancel the common factor of $3$ .

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Step 5.2.2.13.1

Cancel the common factor.

$76 - (0 + \frac{3}{4} \cdot 0^{3 (\frac{4}{3})})$

Step 5.2.2.13.2

Rewrite the expression.

$76 - (0 + \frac{3}{4} \cdot 0^{4})$

Step 5.2.2.14

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

$76 - (0 + \frac{3}{4} \cdot 0)$

Step 5.2.2.15

Multiply $\frac{3}{4}$ by $0$ .

$76 - (0 + 0)$

Step 5.2.2.16

Add $0$ and $0$ .

$76 - 0$

Step 5.2.2.17

Multiply $- 1$ by $0$ .

$76 + 0$

Step 5.2.2.18

Add $76$ and $0$ .

$76$

Step 6