Calculus Examples

$\int_{0}^{8} 2 p \cdot ((0.5 x) \sqrt{1 + {(0.5)}^{2}}) d x$

Step 1

Simplify.

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Step 1.1

Raise $0.5$ to the power of $2$ .

$\int_{0}^{8} 2 p \cdot (0.5 x \sqrt{1 + 0.25}) d x$

Step 1.2

Add $1$ and $0.25$ .

$\int_{0}^{8} 2 p \cdot (0.5 x \sqrt{1.25}) d x$

Step 1.3

Multiply $\sqrt{1.25}$ by $0.5$ .

$\int_{0}^{8} 2 p \cdot (0.55901699 x) d x$

Step 1.4

Multiply $0.55901699$ by $2$ .

$\int_{0}^{8} 1.11803398 p x d x$

Step 2

Since $1.11803398 p$ is constant with respect to $x$ , move $1.11803398 p$ out of the integral.

$1.11803398 p \int_{0}^{8} x d x$

Step 3

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

$1.11803398 p \frac{1}{2} x^{2}]_{0}^{8}$

Step 4

Simplify the answer.

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Step 4.1

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

$1.11803398 p \frac{x^{2}}{2}]_{0}^{8}$

Step 4.2

Substitute and simplify.

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Step 4.2.1

Evaluate $\frac{x^{2}}{2}$ at $8$ and at $0$ .

$1.11803398 p ((\frac{8^{2}}{2}) - \frac{0^{2}}{2})$

Step 4.2.2

Simplify.

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Step 4.2.2.1

Raise $8$ to the power of $2$ .

$1.11803398 p (\frac{64}{2} - \frac{0^{2}}{2})$

Step 4.2.2.2

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

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Step 4.2.2.2.1

Factor $2$ out of $64$ .

$1.11803398 p (\frac{2 \cdot 32}{2} - \frac{0^{2}}{2})$

Step 4.2.2.2.2

Cancel the common factors.

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Step 4.2.2.2.2.1

Factor $2$ out of $2$ .

$1.11803398 p (\frac{2 \cdot 32}{2 (1)} - \frac{0^{2}}{2})$

Step 4.2.2.2.2.2

Cancel the common factor.

$1.11803398 p (\frac{2 \cdot 32}{2 \cdot 1} - \frac{0^{2}}{2})$

Step 4.2.2.2.2.3

Rewrite the expression.

$1.11803398 p (\frac{32}{1} - \frac{0^{2}}{2})$

Step 4.2.2.2.2.4

Divide $32$ by $1$ .

$1.11803398 p (32 - \frac{0^{2}}{2})$

Step 4.2.2.3

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

$1.11803398 p (32 - \frac{0}{2})$

Step 4.2.2.4

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

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Step 4.2.2.4.1

Factor $2$ out of $0$ .

$1.11803398 p (32 - \frac{2 (0)}{2})$

Step 4.2.2.4.2

Cancel the common factors.

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Step 4.2.2.4.2.1

Factor $2$ out of $2$ .

$1.11803398 p (32 - \frac{2 \cdot 0}{2 \cdot 1})$

Step 4.2.2.4.2.2

Cancel the common factor.

$1.11803398 p (32 - \frac{2 \cdot 0}{2 \cdot 1})$

Step 4.2.2.4.2.3

Rewrite the expression.

$1.11803398 p (32 - \frac{0}{1})$

Step 4.2.2.4.2.4

Divide $0$ by $1$ .

$1.11803398 p (32 - 0)$

Step 4.2.2.5

Multiply $- 1$ by $0$ .

$1.11803398 p (32 + 0)$

Step 4.2.2.6

Add $32$ and $0$ .

$1.11803398 p \cdot 32$

Step 4.2.2.7

Multiply $32$ by $1.11803398$ .

$35.77708763 p$

Step 5