Calculus Examples

$- \frac{1}{3} \int_{5}^{2} (x^{3} - 2) d x$

Step 1

Remove parentheses.

$- \frac{1}{3} \cdot \int_{5}^{2} x^{3} - 2 d x$

Step 2

Split the single integral into multiple integrals.

$- \frac{1}{3} (\int_{5}^{2} x^{3} d x + \int_{5}^{2} - 2 d x)$

Step 3

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

$- \frac{1}{3} (\frac{1}{4} x^{4}]_{5}^{2} + \int_{5}^{2} - 2 d x)$

Step 4

Apply the constant rule.

$- \frac{1}{3} (\frac{1}{4} x^{4}]_{5}^{2} + - 2 x]_{5}^{2})$

Step 5

Simplify the answer.

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Step 5.1

Combine $\frac{1}{4} x^{4}]_{5}^{2}$ and $- 2 x]_{5}^{2}$ .

$- \frac{1}{3} \frac{1}{4} x^{4} - 2 x]_{5}^{2}$

Step 5.2

Substitute and simplify.

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Step 5.2.1

Evaluate $\frac{1}{4} x^{4} - 2 x$ at $2$ and at $5$ .

$- \frac{1}{3} ((\frac{1}{4} \cdot 2^{4} - 2 \cdot 2) - (\frac{1}{4} \cdot 5^{4} - 2 \cdot 5))$

Step 5.2.2

Simplify.

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Step 5.2.2.1

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

$- \frac{1}{3} (\frac{1}{4} \cdot 16 - 2 \cdot 2 - (\frac{1}{4} \cdot 5^{4} - 2 \cdot 5))$

Step 5.2.2.2

Combine $\frac{1}{4}$ and $16$ .

$- \frac{1}{3} (\frac{16}{4} - 2 \cdot 2 - (\frac{1}{4} \cdot 5^{4} - 2 \cdot 5))$

Step 5.2.2.3

Cancel the common factor of $16$ and $4$ .

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Step 5.2.2.3.1

Factor $4$ out of $16$ .

$- \frac{1}{3} (\frac{4 \cdot 4}{4} - 2 \cdot 2 - (\frac{1}{4} \cdot 5^{4} - 2 \cdot 5))$

Step 5.2.2.3.2

Cancel the common factors.

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Step 5.2.2.3.2.1

Factor $4$ out of $4$ .

$- \frac{1}{3} (\frac{4 \cdot 4}{4 (1)} - 2 \cdot 2 - (\frac{1}{4} \cdot 5^{4} - 2 \cdot 5))$

Step 5.2.2.3.2.2

Cancel the common factor.

$- \frac{1}{3} (\frac{4 \cdot 4}{4 \cdot 1} - 2 \cdot 2 - (\frac{1}{4} \cdot 5^{4} - 2 \cdot 5))$

Step 5.2.2.3.2.3

Rewrite the expression.

$- \frac{1}{3} (\frac{4}{1} - 2 \cdot 2 - (\frac{1}{4} \cdot 5^{4} - 2 \cdot 5))$

Step 5.2.2.3.2.4

Divide $4$ by $1$ .

$- \frac{1}{3} (4 - 2 \cdot 2 - (\frac{1}{4} \cdot 5^{4} - 2 \cdot 5))$

Step 5.2.2.4

Multiply $- 2$ by $2$ .

$- \frac{1}{3} (4 - 4 - (\frac{1}{4} \cdot 5^{4} - 2 \cdot 5))$

Step 5.2.2.5

Subtract $4$ from $4$ .

$- \frac{1}{3} (0 - (\frac{1}{4} \cdot 5^{4} - 2 \cdot 5))$

Step 5.2.2.6

Raise $5$ to the power of $4$ .

$- \frac{1}{3} (0 - (\frac{1}{4} \cdot 625 - 2 \cdot 5))$

Step 5.2.2.7

Combine $\frac{1}{4}$ and $625$ .

$- \frac{1}{3} (0 - (\frac{625}{4} - 2 \cdot 5))$

Step 5.2.2.8

Multiply $- 2$ by $5$ .

$- \frac{1}{3} (0 - (\frac{625}{4} - 10))$

Step 5.2.2.9

To write $- 10$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$- \frac{1}{3} (0 - (\frac{625}{4} - 10 \cdot \frac{4}{4}))$

Step 5.2.2.10

Combine $- 10$ and $\frac{4}{4}$ .

$- \frac{1}{3} (0 - (\frac{625}{4} + \frac{- 10 \cdot 4}{4}))$

Step 5.2.2.11

Combine the numerators over the common denominator.

$- \frac{1}{3} (0 - \frac{625 - 10 \cdot 4}{4})$

Step 5.2.2.12

Simplify the numerator.

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Step 5.2.2.12.1

Multiply $- 10$ by $4$ .

$- \frac{1}{3} (0 - \frac{625 - 40}{4})$

Step 5.2.2.12.2

Subtract $40$ from $625$ .

$- \frac{1}{3} (0 - \frac{585}{4})$

Step 5.2.2.13

Subtract $\frac{585}{4}$ from $0$ .

$- \frac{1}{3} (- \frac{585}{4})$

Step 5.2.2.14

Multiply $- 1$ by $- 1$ .

$1 (\frac{1}{3}) \frac{585}{4}$

Step 5.2.2.15

Multiply $\frac{1}{3}$ by $1$ .

$\frac{1}{3} \cdot \frac{585}{4}$

Step 5.2.2.16

Multiply $\frac{1}{3}$ by $\frac{585}{4}$ .

$\frac{585}{3 \cdot 4}$

Step 5.2.2.17

Multiply $3$ by $4$ .

$\frac{585}{12}$

Step 5.2.2.18

Cancel the common factor of $585$ and $12$ .

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Step 5.2.2.18.1

Factor $3$ out of $585$ .

$\frac{3 (195)}{12}$

Step 5.2.2.18.2

Cancel the common factors.

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Step 5.2.2.18.2.1

Factor $3$ out of $12$ .

$\frac{3 \cdot 195}{3 \cdot 4}$

Step 5.2.2.18.2.2

Cancel the common factor.

$\frac{3 \cdot 195}{3 \cdot 4}$

Step 5.2.2.18.2.3

Rewrite the expression.

$\frac{195}{4}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$\frac{195}{4}$

Decimal Form:

$48.75$

Mixed Number Form:

$48 \frac{3}{4}$

Step 7