Calculus Examples

$\int_{0}^{b} 9 x^{8} d x$

Step 1

Since $9$ is constant with respect to $x$ , move $9$ out of the integral.

$9 \int_{0}^{b} x^{8} d x$

Step 2

By the Power Rule, the integral of $x^{8}$ with respect to $x$ is $\frac{1}{9} x^{9}$ .

$9 (\frac{1}{9} x^{9}]_{0}^{b})$

Step 3

Simplify the answer.

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Step 3.1

Combine $\frac{1}{9}$ and $x^{9}$ .

$9 (\frac{x^{9}}{9}]_{0}^{b})$

Step 3.2

Substitute and simplify.

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Step 3.2.1

Evaluate $\frac{x^{9}}{9}$ at $b$ and at $0$ .

$9 ((\frac{b^{9}}{9}) - \frac{0^{9}}{9})$

Step 3.2.2

Simplify.

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Step 3.2.2.1

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

$9 (\frac{b^{9}}{9} - \frac{0}{9})$

Step 3.2.2.2

Cancel the common factor of $0$ and $9$ .

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Step 3.2.2.2.1

Factor $9$ out of $0$ .

$9 (\frac{b^{9}}{9} - \frac{9 (0)}{9})$

Step 3.2.2.2.2

Cancel the common factors.

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Step 3.2.2.2.2.1

Factor $9$ out of $9$ .

$9 (\frac{b^{9}}{9} - \frac{9 \cdot 0}{9 \cdot 1})$

Step 3.2.2.2.2.2

Cancel the common factor.

$9 (\frac{b^{9}}{9} - \frac{9 \cdot 0}{9 \cdot 1})$

Step 3.2.2.2.2.3

Rewrite the expression.

$9 (\frac{b^{9}}{9} - \frac{0}{1})$

Step 3.2.2.2.2.4

Divide $0$ by $1$ .

$9 (\frac{b^{9}}{9} - 0)$

Step 3.2.2.3

Multiply $- 1$ by $0$ .

$9 (\frac{b^{9}}{9} + 0)$

Step 3.2.2.4

Add $\frac{b^{9}}{9}$ and $0$ .

$9 \frac{b^{9}}{9}$

Step 3.2.2.5

Combine $9$ and $\frac{b^{9}}{9}$ .

$\frac{9 b^{9}}{9}$

Step 3.2.2.6

Cancel the common factor of $9$ .

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Step 3.2.2.6.1

Cancel the common factor.

$\frac{9 b^{9}}{9}$

Step 3.2.2.6.2

Divide $b^{9}$ by $1$ .

$b^{9}$