Calculus Examples

$\int_{3}^{\sqrt{13}} x d x$

Step 1

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

$\frac{1}{2} x^{2}]_{3}^{\sqrt{13}}$

Step 2

Substitute and simplify.

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Step 2.1

Evaluate $\frac{1}{2} x^{2}$ at $\sqrt{13}$ and at $3$ .

$(\frac{1}{2} {\sqrt{13}}^{2}) - \frac{1}{2} \cdot 3^{2}$

Step 2.2

Simplify.

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Step 2.2.1

Rewrite ${\sqrt{13}}^{2}$ as $13$ .

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Step 2.2.1.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{13}$ as $13^{\frac{1}{2}}$ .

$\frac{1}{2} {(13^{\frac{1}{2}})}^{2} - \frac{1}{2} \cdot 3^{2}$

Step 2.2.1.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\frac{1}{2} \cdot 13^{\frac{1}{2} \cdot 2} - \frac{1}{2} \cdot 3^{2}$

Step 2.2.1.3

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

$\frac{1}{2} \cdot 13^{\frac{2}{2}} - \frac{1}{2} \cdot 3^{2}$

Step 2.2.1.4

Cancel the common factor of $2$ .

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Step 2.2.1.4.1

Cancel the common factor.

$\frac{1}{2} \cdot 13^{\frac{2}{2}} - \frac{1}{2} \cdot 3^{2}$

Step 2.2.1.4.2

Rewrite the expression.

$\frac{1}{2} \cdot 13^{1} - \frac{1}{2} \cdot 3^{2}$

Step 2.2.1.5

Evaluate the exponent.

$\frac{1}{2} \cdot 13 - \frac{1}{2} \cdot 3^{2}$

Step 2.2.2

Combine $\frac{1}{2}$ and $13$ .

$\frac{13}{2} - \frac{1}{2} \cdot 3^{2}$

Step 2.2.3

Raise $3$ to the power of $2$ .

$\frac{13}{2} - \frac{1}{2} \cdot 9$

Step 2.2.4

Multiply $9$ by $- 1$ .

$\frac{13}{2} - 9 (\frac{1}{2})$

Step 2.2.5

Combine $- 9$ and $\frac{1}{2}$ .

$\frac{13}{2} + \frac{- 9}{2}$

Step 2.2.6

Move the negative in front of the fraction.

$\frac{13}{2} - \frac{9}{2}$

Step 2.2.7

Combine the numerators over the common denominator.

$\frac{13 - 9}{2}$

Step 2.2.8

Subtract $9$ from $13$ .

$\frac{4}{2}$

Step 2.2.9

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

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Step 2.2.9.1

Factor $2$ out of $4$ .

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

Step 2.2.9.2

Cancel the common factors.

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Step 2.2.9.2.1

Factor $2$ out of $2$ .

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

Step 2.2.9.2.2

Cancel the common factor.

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

Step 2.2.9.2.3

Rewrite the expression.

$\frac{2}{1}$

Step 2.2.9.2.4

Divide $2$ by $1$ .

$2$

Step 3