Calculus Examples

$\int_{0}^{4} 8 - 2 x d x$

Step 1

Split the single integral into multiple integrals.

$\int_{0}^{4} 8 d x + \int_{0}^{4} - 2 x d x$

Step 2

Apply the constant rule.

$8 x]_{0}^{4} + \int_{0}^{4} - 2 x d x$

Step 3

Since $- 2$ is constant with respect to $x$ , move $- 2$ out of the integral.

$8 x]_{0}^{4} - 2 \int_{0}^{4} x d x$

Step 4

By the Power Rule, the integral of $x$ with respect to $x$ is $\frac{1}{2} x^{2}$ .

$8 x]_{0}^{4} - 2 (\frac{1}{2} x^{2}]_{0}^{4})$

Step 5

Simplify the answer.

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Step 5.1

Combine $\frac{1}{2}$ and $x^{2}$ .

$8 x]_{0}^{4} - 2 (\frac{x^{2}}{2}]_{0}^{4})$

Step 5.2

Substitute and simplify.

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Step 5.2.1

Evaluate $8 x$ at $4$ and at $0$ .

$(8 \cdot 4) - 8 \cdot 0 - 2 (\frac{x^{2}}{2}]_{0}^{4})$

Step 5.2.2

Evaluate $\frac{x^{2}}{2}$ at $4$ and at $0$ .

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

Step 5.2.3

Simplify.

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Step 5.2.3.1

Multiply $8$ by $4$ .

$32 - 8 \cdot 0 - 2 (\frac{4^{2}}{2} - \frac{0^{2}}{2})$

Step 5.2.3.2

Multiply $- 8$ by $0$ .

$32 + 0 - 2 (\frac{4^{2}}{2} - \frac{0^{2}}{2})$

Step 5.2.3.3

Add $32$ and $0$ .

$32 - 2 (\frac{4^{2}}{2} - \frac{0^{2}}{2})$

Step 5.2.3.4

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

$32 - 2 (\frac{16}{2} - \frac{0^{2}}{2})$

Step 5.2.3.5

Cancel the common factor of $16$ and $2$ .

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Step 5.2.3.5.1

Factor $2$ out of $16$ .

$32 - 2 (\frac{2 \cdot 8}{2} - \frac{0^{2}}{2})$

Step 5.2.3.5.2

Cancel the common factors.

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Step 5.2.3.5.2.1

Factor $2$ out of $2$ .

$32 - 2 (\frac{2 \cdot 8}{2 (1)} - \frac{0^{2}}{2})$

Step 5.2.3.5.2.2

Cancel the common factor.

$32 - 2 (\frac{2 \cdot 8}{2 \cdot 1} - \frac{0^{2}}{2})$

Step 5.2.3.5.2.3

Rewrite the expression.

$32 - 2 (\frac{8}{1} - \frac{0^{2}}{2})$

Step 5.2.3.5.2.4

Divide $8$ by $1$ .

$32 - 2 (8 - \frac{0^{2}}{2})$

Step 5.2.3.6

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

$32 - 2 (8 - \frac{0}{2})$

Step 5.2.3.7

Cancel the common factor of $0$ and $2$ .

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Step 5.2.3.7.1

Factor $2$ out of $0$ .

$32 - 2 (8 - \frac{2 (0)}{2})$

Step 5.2.3.7.2

Cancel the common factors.

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Step 5.2.3.7.2.1

Factor $2$ out of $2$ .

$32 - 2 (8 - \frac{2 \cdot 0}{2 \cdot 1})$

Step 5.2.3.7.2.2

Cancel the common factor.

$32 - 2 (8 - \frac{2 \cdot 0}{2 \cdot 1})$

Step 5.2.3.7.2.3

Rewrite the expression.

$32 - 2 (8 - \frac{0}{1})$

Step 5.2.3.7.2.4

Divide $0$ by $1$ .

$32 - 2 (8 - 0)$

Step 5.2.3.8

Multiply $- 1$ by $0$ .

$32 - 2 (8 + 0)$

Step 5.2.3.9

Add $8$ and $0$ .

$32 - 2 \cdot 8$

Step 5.2.3.10

Multiply $- 2$ by $8$ .

$32 - 16$

Step 5.2.3.11

Subtract $16$ from $32$ .

$16$

Step 6