Calculus Examples

$\int_{0}^{4} 24 - \frac{6 x^{3}}{12} d x$

Step 1

Cancel the common factor of $6$ and $12$ .

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Step 1.1

Factor $6$ out of $6 x^{3}$ .

$\int_{0}^{4} 24 - \frac{6 (x^{3})}{12} d x$

Step 1.2

Cancel the common factors.

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Step 1.2.1

Factor $6$ out of $12$ .

$\int_{0}^{4} 24 - \frac{6 x^{3}}{6 \cdot 2} d x$

Step 1.2.2

Cancel the common factor.

$\int_{0}^{4} 24 - \frac{6 x^{3}}{6 \cdot 2} d x$

Step 1.2.3

Rewrite the expression.

$\int_{0}^{4} 24 - \frac{x^{3}}{2} d x$

Step 2

Split the single integral into multiple integrals.

$\int_{0}^{4} 24 d x + \int_{0}^{4} - \frac{x^{3}}{2} d x$

Step 3

Apply the constant rule.

$24 x]_{0}^{4} + \int_{0}^{4} - \frac{x^{3}}{2} d x$

Step 4

Since $- 1$ is constant with respect to $x$ , move $- 1$ out of the integral.

$24 x]_{0}^{4} - \int_{0}^{4} \frac{x^{3}}{2} d x$

Step 5

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

$24 x]_{0}^{4} - (\frac{1}{2} \int_{0}^{4} x^{3} d x)$

Step 6

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

$24 x]_{0}^{4} - \frac{1}{2} \frac{1}{4} x^{4}]_{0}^{4}$

Step 7

Substitute and simplify.

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Step 7.1

Evaluate $24 x$ at $4$ and at $0$ .

$(24 \cdot 4) - 24 \cdot 0 - \frac{1}{2} \frac{1}{4} x^{4}]_{0}^{4}$

Step 7.2

Evaluate $\frac{1}{4} x^{4}$ at $4$ and at $0$ .

$24 \cdot 4 - 24 \cdot 0 - \frac{1}{2} (\frac{1}{4} \cdot 4^{4} - \frac{1}{4} \cdot 0^{4})$

Step 7.3

Simplify.

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Step 7.3.1

Multiply $24$ by $4$ .

$96 - 24 \cdot 0 - \frac{1}{2} (\frac{1}{4} \cdot 4^{4} - \frac{1}{4} \cdot 0^{4})$

Step 7.3.2

Multiply $- 24$ by $0$ .

$96 + 0 - \frac{1}{2} (\frac{1}{4} \cdot 4^{4} - \frac{1}{4} \cdot 0^{4})$

Step 7.3.3

Add $96$ and $0$ .

$96 - \frac{1}{2} (\frac{1}{4} \cdot 4^{4} - \frac{1}{4} \cdot 0^{4})$

Step 7.3.4

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

$96 - \frac{1}{2} (\frac{1}{4} \cdot 256 - \frac{1}{4} \cdot 0^{4})$

Step 7.3.5

Combine $\frac{1}{4}$ and $256$ .

$96 - \frac{1}{2} (\frac{256}{4} - \frac{1}{4} \cdot 0^{4})$

Step 7.3.6

Cancel the common factor of $256$ and $4$ .

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Step 7.3.6.1

Factor $4$ out of $256$ .

$96 - \frac{1}{2} (\frac{4 \cdot 64}{4} - \frac{1}{4} \cdot 0^{4})$

Step 7.3.6.2

Cancel the common factors.

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Step 7.3.6.2.1

Factor $4$ out of $4$ .

$96 - \frac{1}{2} (\frac{4 \cdot 64}{4 (1)} - \frac{1}{4} \cdot 0^{4})$

Step 7.3.6.2.2

Cancel the common factor.

$96 - \frac{1}{2} (\frac{4 \cdot 64}{4 \cdot 1} - \frac{1}{4} \cdot 0^{4})$

Step 7.3.6.2.3

Rewrite the expression.

$96 - \frac{1}{2} (\frac{64}{1} - \frac{1}{4} \cdot 0^{4})$

Step 7.3.6.2.4

Divide $64$ by $1$ .

$96 - \frac{1}{2} (64 - \frac{1}{4} \cdot 0^{4})$

Step 7.3.7

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

$96 - \frac{1}{2} (64 - \frac{1}{4} \cdot 0)$

Step 7.3.8

Multiply $0$ by $- 1$ .

$96 - \frac{1}{2} (64 + 0 (\frac{1}{4}))$

Step 7.3.9

Multiply $0$ by $\frac{1}{4}$ .

$96 - \frac{1}{2} (64 + 0)$

Step 7.3.10

Add $64$ and $0$ .

$96 - \frac{1}{2} \cdot 64$

Step 7.3.11

Multiply $64$ by $- 1$ .

$96 - 64 (\frac{1}{2})$

Step 7.3.12

Combine $- 64$ and $\frac{1}{2}$ .

$96 + \frac{- 64}{2}$

Step 7.3.13

Cancel the common factor of $- 64$ and $2$ .

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Step 7.3.13.1

Factor $2$ out of $- 64$ .

$96 + \frac{2 \cdot - 32}{2}$

Step 7.3.13.2

Cancel the common factors.

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Step 7.3.13.2.1

Factor $2$ out of $2$ .

$96 + \frac{2 \cdot - 32}{2 (1)}$

Step 7.3.13.2.2

Cancel the common factor.

$96 + \frac{2 \cdot - 32}{2 \cdot 1}$

Step 7.3.13.2.3

Rewrite the expression.

$96 + \frac{- 32}{1}$

Step 7.3.13.2.4

Divide $- 32$ by $1$ .

$96 - 32$

Step 7.3.14

Subtract $32$ from $96$ .

$64$

Step 8