Calculus Examples

$\int_{0.8}^{2.2} \sqrt[3]{x} - 2 d x$

Step 1

Split the single integral into multiple integrals.

$\int_{0.8}^{2.2} \sqrt[3]{x} d x + \int_{0.8}^{2.2} - 2 d x$

Step 2

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

$\int_{0.8}^{2.2} x^{\frac{1}{3}} d x + \int_{0.8}^{2.2} - 2 d x$

Step 3

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

$\frac{3}{4} x^{\frac{4}{3}}]_{0.8}^{2.2} + \int_{0.8}^{2.2} - 2 d x$

Step 4

Apply the constant rule.

$\frac{3}{4} x^{\frac{4}{3}}]_{0.8}^{2.2} + - 2 x]_{0.8}^{2.2}$

Step 5

Simplify the answer.

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Step 5.1

Combine $\frac{3}{4} x^{\frac{4}{3}}]_{0.8}^{2.2}$ and $- 2 x]_{0.8}^{2.2}$ .

$\frac{3}{4} x^{\frac{4}{3}} - 2 x]_{0.8}^{2.2}$

Step 5.2

Substitute and simplify.

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Step 5.2.1

Evaluate $\frac{3}{4} x^{\frac{4}{3}} - 2 x$ at $2.2$ and at $0.8$ .

$(\frac{3}{4} \cdot {2.2}^{\frac{4}{3}} - 2 \cdot 2.2) - (\frac{3}{4} \cdot {0.8}^{\frac{4}{3}} - 2 \cdot 0.8)$

Step 5.2.2

Simplify.

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Step 5.2.2.1

Raise $2.2$ to the power of $\frac{4}{3}$ .

$\frac{3}{4} \cdot 2.86130118 - 2 \cdot 2.2 - (\frac{3}{4} \cdot {0.8}^{\frac{4}{3}} - 2 \cdot 0.8)$

Step 5.2.2.2

Combine $\frac{3}{4}$ and $2.86130118$ .

$\frac{3 \cdot 2.86130118}{4} - 2 \cdot 2.2 - (\frac{3}{4} \cdot {0.8}^{\frac{4}{3}} - 2 \cdot 0.8)$

Step 5.2.2.3

Multiply $3$ by $2.86130118$ .

$\frac{8.58390354}{4} - 2 \cdot 2.2 - (\frac{3}{4} \cdot {0.8}^{\frac{4}{3}} - 2 \cdot 0.8)$

Step 5.2.2.4

Multiply $- 2$ by $2.2$ .

$\frac{8.58390354}{4} - 4.4 - (\frac{3}{4} \cdot {0.8}^{\frac{4}{3}} - 2 \cdot 0.8)$

Step 5.2.2.5

To write $- 4.4$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$\frac{8.58390354}{4} - 4.4 \cdot \frac{4}{4} - (\frac{3}{4} \cdot {0.8}^{\frac{4}{3}} - 2 \cdot 0.8)$

Step 5.2.2.6

Combine $- 4.4$ and $\frac{4}{4}$ .

$\frac{8.58390354}{4} + \frac{- 4.4 \cdot 4}{4} - (\frac{3}{4} \cdot {0.8}^{\frac{4}{3}} - 2 \cdot 0.8)$

Step 5.2.2.7

Combine the numerators over the common denominator.

$\frac{8.58390354 - 4.4 \cdot 4}{4} - (\frac{3}{4} \cdot {0.8}^{\frac{4}{3}} - 2 \cdot 0.8)$

Step 5.2.2.8

Multiply $- 4.4$ by $4$ .

$\frac{8.58390354 - 17.6}{4} - (\frac{3}{4} \cdot {0.8}^{\frac{4}{3}} - 2 \cdot 0.8)$

Step 5.2.2.9

Subtract $17.6$ from $8.58390354$ .

$\frac{- 9.01609645}{4} - (\frac{3}{4} \cdot {0.8}^{\frac{4}{3}} - 2 \cdot 0.8)$

Step 5.2.2.10

Move the negative in front of the fraction.

$- \frac{9.01609645}{4} - (\frac{3}{4} \cdot {0.8}^{\frac{4}{3}} - 2 \cdot 0.8)$

Step 5.2.2.11

Raise $0.8$ to the power of $\frac{4}{3}$ .

$- \frac{9.01609645}{4} - (\frac{3}{4} \cdot 0.74265421 - 2 \cdot 0.8)$

Step 5.2.2.12

Combine $\frac{3}{4}$ and $0.74265421$ .

$- \frac{9.01609645}{4} - (\frac{3 \cdot 0.74265421}{4} - 2 \cdot 0.8)$

Step 5.2.2.13

Multiply $3$ by $0.74265421$ .

$- \frac{9.01609645}{4} - (\frac{2.22796264}{4} - 2 \cdot 0.8)$

Step 5.2.2.14

Multiply $- 2$ by $0.8$ .

$- \frac{9.01609645}{4} - (\frac{2.22796264}{4} - 1.6)$

Step 5.2.2.15

To write $- 1.6$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$- \frac{9.01609645}{4} - (\frac{2.22796264}{4} - 1.6 \cdot \frac{4}{4})$

Step 5.2.2.16

Combine $- 1.6$ and $\frac{4}{4}$ .

$- \frac{9.01609645}{4} - (\frac{2.22796264}{4} + \frac{- 1.6 \cdot 4}{4})$

Step 5.2.2.17

Combine the numerators over the common denominator.

$- \frac{9.01609645}{4} - \frac{2.22796264 - 1.6 \cdot 4}{4}$

Step 5.2.2.18

Multiply $- 1.6$ by $4$ .

$- \frac{9.01609645}{4} - \frac{2.22796264 - 6.4}{4}$

Step 5.2.2.19

Subtract $6.4$ from $2.22796264$ .

$- \frac{9.01609645}{4} - \frac{- 4.17203735}{4}$

Step 5.2.2.20

Move the negative in front of the fraction.

$- \frac{9.01609645}{4} - - \frac{4.17203735}{4}$

Step 5.2.2.21

Multiply $- 1$ by $- 1$ .

$- \frac{9.01609645}{4} + 1 (\frac{4.17203735}{4})$

Step 5.2.2.22

Multiply $\frac{4.17203735}{4}$ by $1$ .

$- \frac{9.01609645}{4} + \frac{4.17203735}{4}$

Step 5.2.2.23

Combine the numerators over the common denominator.

$\frac{- 9.01609645 + 4.17203735}{4}$

Step 5.2.2.24

Add $- 9.01609645$ and $4.17203735$ .

$\frac{- 4.84405909}{4}$

Step 5.2.2.25

Move the negative in front of the fraction.

$- \frac{4.84405909}{4}$

Step 6

Simplify.

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Step 6.1

Divide $4.84405909$ by $4$ .

$- 1 \cdot 1.21101477$

Step 6.2

Multiply $- 1$ by $1.21101477$ .

$- 1.21101477$

Step 7