Calculus Examples

$\int_{7}^{2} g (x) d x$

Step 1

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

$g \int_{7}^{2} x d x$

Step 2

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

$g \frac{1}{2} x^{2}]_{7}^{2}$

Step 3

Simplify the answer.

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Step 3.1

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

$g \frac{x^{2}}{2}]_{7}^{2}$

Step 3.2

Substitute and simplify.

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Step 3.2.1

Evaluate $\frac{x^{2}}{2}$ at $2$ and at $7$ .

$g ((\frac{2^{2}}{2}) - \frac{7^{2}}{2})$

Step 3.2.2

Simplify.

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Step 3.2.2.1

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

$g (\frac{4}{2} - \frac{7^{2}}{2})$

Step 3.2.2.2

Cancel the common factor of $4$ and $2$ .

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Step 3.2.2.2.1

Factor $2$ out of $4$ .

$g (\frac{2 \cdot 2}{2} - \frac{7^{2}}{2})$

Step 3.2.2.2.2

Cancel the common factors.

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Step 3.2.2.2.2.1

Factor $2$ out of $2$ .

$g (\frac{2 \cdot 2}{2 (1)} - \frac{7^{2}}{2})$

Step 3.2.2.2.2.2

Cancel the common factor.

$g (\frac{2 \cdot 2}{2 \cdot 1} - \frac{7^{2}}{2})$

Step 3.2.2.2.2.3

Rewrite the expression.

$g (\frac{2}{1} - \frac{7^{2}}{2})$

Step 3.2.2.2.2.4

Divide $2$ by $1$ .

$g (2 - \frac{7^{2}}{2})$

Step 3.2.2.3

Raise $7$ to the power of $2$ .

$g (2 - \frac{49}{2})$

Step 3.2.2.4

To write $2$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$g (2 \cdot \frac{2}{2} - \frac{49}{2})$

Step 3.2.2.5

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

$g (\frac{2 \cdot 2}{2} - \frac{49}{2})$

Step 3.2.2.6

Combine the numerators over the common denominator.

$g \frac{2 \cdot 2 - 49}{2}$

Step 3.2.2.7

Simplify the numerator.

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Step 3.2.2.7.1

Multiply $2$ by $2$ .

$g \frac{4 - 49}{2}$

Step 3.2.2.7.2

Subtract $49$ from $4$ .

$g \frac{- 45}{2}$

Step 3.2.2.8

Move the negative in front of the fraction.

$g (- \frac{45}{2})$

Step 3.2.2.9

Combine $g$ and $\frac{45}{2}$ .

$- \frac{g \cdot 45}{2}$

Step 3.2.2.10

Move $45$ to the left of $g$ .

$- \frac{45 g}{2}$

Step 3.3

Reorder terms.

$- \frac{45}{2} g$

Step 4

Simplify.

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Step 4.1

Combine $g$ and $\frac{45}{2}$ .

$- \frac{g \cdot 45}{2}$

Step 4.2

Move $45$ to the left of $g$ .

$- \frac{45 g}{2}$

Step 5