Calculus Examples

$\int_{2}^{5} x d x$

Step 1

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

$\frac{1}{2} x^{2}]_{2}^{5}$

Step 2

Substitute and simplify.

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Step 2.1

Evaluate $\frac{1}{2} x^{2}$ at $5$ and at $2$ .

$(\frac{1}{2} \cdot 5^{2}) - \frac{1}{2} \cdot 2^{2}$

Step 2.2

Simplify.

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Step 2.2.1

Raise $5$ to the power of $2$ .

$\frac{1}{2} \cdot 25 - \frac{1}{2} \cdot 2^{2}$

Step 2.2.2

Combine $\frac{1}{2}$ and $25$ .

$\frac{25}{2} - \frac{1}{2} \cdot 2^{2}$

Step 2.2.3

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

$\frac{25}{2} - \frac{1}{2} \cdot 4$

Step 2.2.4

Multiply $4$ by $- 1$ .

$\frac{25}{2} - 4 (\frac{1}{2})$

Step 2.2.5

Combine $- 4$ and $\frac{1}{2}$ .

$\frac{25}{2} + \frac{- 4}{2}$

Step 2.2.6

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

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Step 2.2.6.1

Factor $2$ out of $- 4$ .

$\frac{25}{2} + \frac{2 \cdot - 2}{2}$

Step 2.2.6.2

Cancel the common factors.

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Step 2.2.6.2.1

Factor $2$ out of $2$ .

$\frac{25}{2} + \frac{2 \cdot - 2}{2 (1)}$

Step 2.2.6.2.2

Cancel the common factor.

$\frac{25}{2} + \frac{2 \cdot - 2}{2 \cdot 1}$

Step 2.2.6.2.3

Rewrite the expression.

$\frac{25}{2} + \frac{- 2}{1}$

Step 2.2.6.2.4

Divide $- 2$ by $1$ .

$\frac{25}{2} - 2$

Step 2.2.7

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

$\frac{25}{2} - 2 \cdot \frac{2}{2}$

Step 2.2.8

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

$\frac{25}{2} + \frac{- 2 \cdot 2}{2}$

Step 2.2.9

Combine the numerators over the common denominator.

$\frac{25 - 2 \cdot 2}{2}$

Step 2.2.10

Simplify the numerator.

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Step 2.2.10.1

Multiply $- 2$ by $2$ .

$\frac{25 - 4}{2}$

Step 2.2.10.2

Subtract $4$ from $25$ .

$\frac{21}{2}$

Step 3

The result can be shown in multiple forms.

Exact Form:

$\frac{21}{2}$

Decimal Form:

$10.5$

Mixed Number Form:

$10 \frac{1}{2}$

Step 4