Calculus Examples

$\int_{0}^{6} \frac{1}{2} x d x$

Step 1

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

$\frac{1}{2} \int_{0}^{6} x d x$

Step 2

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

$\frac{1}{2} \frac{1}{2} x^{2}]_{0}^{6}$

Step 3

Substitute and simplify.

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Step 3.1

Evaluate $\frac{1}{2} x^{2}$ at $6$ and at $0$ .

$\frac{1}{2} ((\frac{1}{2} \cdot 6^{2}) - \frac{1}{2} \cdot 0^{2})$

Step 3.2

Simplify.

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Step 3.2.1

Raise $6$ to the power of $2$ .

$\frac{1}{2} (\frac{1}{2} \cdot 36 - \frac{1}{2} \cdot 0^{2})$

Step 3.2.2

Combine $\frac{1}{2}$ and $36$ .

$\frac{1}{2} (\frac{36}{2} - \frac{1}{2} \cdot 0^{2})$

Step 3.2.3

Cancel the common factor of $36$ and $2$ .

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Step 3.2.3.1

Factor $2$ out of $36$ .

$\frac{1}{2} (\frac{2 \cdot 18}{2} - \frac{1}{2} \cdot 0^{2})$

Step 3.2.3.2

Cancel the common factors.

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Step 3.2.3.2.1

Factor $2$ out of $2$ .

$\frac{1}{2} (\frac{2 \cdot 18}{2 (1)} - \frac{1}{2} \cdot 0^{2})$

Step 3.2.3.2.2

Cancel the common factor.

$\frac{1}{2} (\frac{2 \cdot 18}{2 \cdot 1} - \frac{1}{2} \cdot 0^{2})$

Step 3.2.3.2.3

Rewrite the expression.

$\frac{1}{2} (\frac{18}{1} - \frac{1}{2} \cdot 0^{2})$

Step 3.2.3.2.4

Divide $18$ by $1$ .

$\frac{1}{2} (18 - \frac{1}{2} \cdot 0^{2})$

Step 3.2.4

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

$\frac{1}{2} (18 - \frac{1}{2} \cdot 0)$

Step 3.2.5

Multiply $0$ by $- 1$ .

$\frac{1}{2} (18 + 0 (\frac{1}{2}))$

Step 3.2.6

Multiply $0$ by $\frac{1}{2}$ .

$\frac{1}{2} (18 + 0)$

Step 3.2.7

Add $18$ and $0$ .

$\frac{1}{2} \cdot 18$

Step 3.2.8

Combine $\frac{1}{2}$ and $18$ .

$\frac{18}{2}$

Step 3.2.9

Cancel the common factor of $18$ and $2$ .

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Step 3.2.9.1

Factor $2$ out of $18$ .

$\frac{2 \cdot 9}{2}$

Step 3.2.9.2

Cancel the common factors.

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Step 3.2.9.2.1

Factor $2$ out of $2$ .

$\frac{2 \cdot 9}{2 (1)}$

Step 3.2.9.2.2

Cancel the common factor.

$\frac{2 \cdot 9}{2 \cdot 1}$

Step 3.2.9.2.3

Rewrite the expression.

$\frac{9}{1}$

Step 3.2.9.2.4

Divide $9$ by $1$ .

$9$

Step 4