Calculus Examples

$\int_{b}^{2 b} x^{9} d x$

Step 1

By the Power Rule, the integral of $x^{9}$ with respect to $x$ is $\frac{1}{10} x^{10}$ .

$\frac{1}{10} x^{10}]_{b}^{2 b}$

Step 2

Simplify the answer.

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Step 2.1

Substitute and simplify.

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Step 2.1.1

Evaluate $\frac{1}{10} x^{10}$ at $2 b$ and at $b$ .

$(\frac{1}{10} {(2 b)}^{10}) - \frac{1}{10} b^{10}$

Step 2.1.2

Simplify.

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Step 2.1.2.1

Factor $2$ out of $2 b$ .

$\frac{1}{10} {(2 (b))}^{10} - \frac{1}{10} b^{10}$

Step 2.1.2.2

Apply the product rule to $2 (b)$ .

$\frac{1}{10} (2^{10} b^{10}) - \frac{1}{10} b^{10}$

Step 2.1.2.3

Raise $2$ to the power of $10$ .

$\frac{1}{10} (1024 b^{10}) - \frac{1}{10} b^{10}$

Step 2.1.2.4

Combine $1024$ and $\frac{1}{10}$ .

$\frac{1024}{10} b^{10} - \frac{1}{10} b^{10}$

Step 2.1.2.5

Combine $\frac{1024}{10}$ and $b^{10}$ .

$\frac{1024 b^{10}}{10} - \frac{1}{10} b^{10}$

Step 2.1.2.6

Cancel the common factor of $1024$ and $10$ .

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Step 2.1.2.6.1

Factor $2$ out of $1024 b^{10}$ .

$\frac{2 (512 b^{10})}{10} - \frac{1}{10} b^{10}$

Step 2.1.2.6.2

Cancel the common factors.

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Step 2.1.2.6.2.1

Factor $2$ out of $10$ .

$\frac{2 (512 b^{10})}{2 (5)} - \frac{1}{10} b^{10}$

Step 2.1.2.6.2.2

Cancel the common factor.

$\frac{2 (512 b^{10})}{2 \cdot 5} - \frac{1}{10} b^{10}$

Step 2.1.2.6.2.3

Rewrite the expression.

$\frac{512 b^{10}}{5} - \frac{1}{10} b^{10}$

Step 2.1.2.7

To write $- \frac{1}{10} b^{10}$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$\frac{512 b^{10}}{5} - \frac{1}{10} b^{10} \cdot \frac{5}{5}$

Step 2.1.2.8

Combine $- \frac{1}{10} b^{10}$ and $\frac{5}{5}$ .

$\frac{512 b^{10}}{5} + \frac{- \frac{1}{10} b^{10} \cdot 5}{5}$

Step 2.1.2.9

Combine the numerators over the common denominator.

$\frac{512 b^{10} - \frac{1}{10} b^{10} \cdot 5}{5}$

Step 2.1.2.10

Combine $b^{10}$ and $\frac{1}{10}$ .

$\frac{512 b^{10} - \frac{b^{10}}{10} \cdot 5}{5}$

Step 2.1.2.11

Multiply $5$ by $- 1$ .

$\frac{512 b^{10} - 5 \frac{b^{10}}{10}}{5}$

Step 2.1.2.12

Combine $- 5$ and $\frac{b^{10}}{10}$ .

$\frac{512 b^{10} + \frac{- 5 b^{10}}{10}}{5}$

Step 2.1.2.13

Cancel the common factor of $- 5$ and $10$ .

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Step 2.1.2.13.1

Factor $5$ out of $- 5 b^{10}$ .

$\frac{512 b^{10} + \frac{5 (- b^{10})}{10}}{5}$

Step 2.1.2.13.2

Cancel the common factors.

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Step 2.1.2.13.2.1

Factor $5$ out of $10$ .

$\frac{512 b^{10} + \frac{5 (- b^{10})}{5 (2)}}{5}$

Step 2.1.2.13.2.2

Cancel the common factor.

$\frac{512 b^{10} + \frac{5 (- b^{10})}{5 \cdot 2}}{5}$

Step 2.1.2.13.2.3

Rewrite the expression.

$\frac{512 b^{10} + \frac{- b^{10}}{2}}{5}$

Step 2.1.2.14

Move the negative in front of the fraction.

$\frac{512 b^{10} - \frac{b^{10}}{2}}{5}$

Step 2.1.2.15

To write $512 b^{10}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{512 b^{10} \cdot \frac{2}{2} - \frac{b^{10}}{2}}{5}$

Step 2.1.2.16

Combine $512 b^{10}$ and $\frac{2}{2}$ .

$\frac{\frac{512 b^{10} \cdot 2}{2} - \frac{b^{10}}{2}}{5}$

Step 2.1.2.17

Combine the numerators over the common denominator.

$\frac{\frac{512 b^{10} \cdot 2 - b^{10}}{2}}{5}$

Step 2.1.2.18

Multiply $2$ by $512$ .

$\frac{\frac{1024 b^{10} - b^{10}}{2}}{5}$

Step 2.1.2.19

Subtract $b^{10}$ from $1024 b^{10}$ .

$\frac{\frac{1023 b^{10}}{2}}{5}$

Step 2.1.2.20

Rewrite $\frac{\frac{1023 b^{10}}{2}}{5}$ as a product.

$\frac{1023 b^{10}}{2} \cdot \frac{1}{5}$

Step 2.1.2.21

Multiply $\frac{1023 b^{10}}{2}$ by $\frac{1}{5}$ .

$\frac{1023 b^{10}}{2 \cdot 5}$

Step 2.1.2.22

Multiply $2$ by $5$ .

$\frac{1023 b^{10}}{10}$

Step 2.2

Reorder terms.

$\frac{1023}{10} b^{10}$

Step 3

Combine $\frac{1023}{10}$ and $b^{10}$ .

$\frac{1023 b^{10}}{10}$