Calculus Examples

$\int_{0}^{1} (1 - x^{9}) d x$

Step 1

Remove parentheses.

$\int_{0}^{1} 1 - x^{9} d x$

Step 2

Split the single integral into multiple integrals.

$\int_{0}^{1} d x + \int_{0}^{1} - x^{9} d x$

Step 3

Apply the constant rule.

$x]_{0}^{1} + \int_{0}^{1} - x^{9} d x$

Step 4

Since $- 1$ is constant with respect to $x$ , move $- 1$ out of the integral.

$x]_{0}^{1} - \int_{0}^{1} x^{9} d x$

Step 5

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

$x]_{0}^{1} - (\frac{1}{10} x^{10}]_{0}^{1})$

Step 6

Simplify the answer.

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Step 6.1

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

$x]_{0}^{1} - (\frac{x^{10}}{10}]_{0}^{1})$

Step 6.2

Substitute and simplify.

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Step 6.2.1

Evaluate $x$ at $1$ and at $0$ .

$(1) + 0 - (\frac{x^{10}}{10}]_{0}^{1})$

Step 6.2.2

Evaluate $\frac{x^{10}}{10}$ at $1$ and at $0$ .

$1 + 0 - (\frac{1^{10}}{10} - \frac{0^{10}}{10})$

Step 6.2.3

Simplify.

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Step 6.2.3.1

Add $1$ and $0$ .

$1 - (\frac{1^{10}}{10} - \frac{0^{10}}{10})$

Step 6.2.3.2

One to any power is one.

$1 - (\frac{1}{10} - \frac{0^{10}}{10})$

Step 6.2.3.3

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

$1 - (\frac{1}{10} - \frac{0}{10})$

Step 6.2.3.4

Cancel the common factor of $0$ and $10$ .

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Step 6.2.3.4.1

Factor $10$ out of $0$ .

$1 - (\frac{1}{10} - \frac{10 (0)}{10})$

Step 6.2.3.4.2

Cancel the common factors.

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Step 6.2.3.4.2.1

Factor $10$ out of $10$ .

$1 - (\frac{1}{10} - \frac{10 \cdot 0}{10 \cdot 1})$

Step 6.2.3.4.2.2

Cancel the common factor.

$1 - (\frac{1}{10} - \frac{10 \cdot 0}{10 \cdot 1})$

Step 6.2.3.4.2.3

Rewrite the expression.

$1 - (\frac{1}{10} - \frac{0}{1})$

Step 6.2.3.4.2.4

Divide $0$ by $1$ .

$1 - (\frac{1}{10} - 0)$

Step 6.2.3.5

Multiply $- 1$ by $0$ .

$1 - (\frac{1}{10} + 0)$

Step 6.2.3.6

Add $\frac{1}{10}$ and $0$ .

$1 - \frac{1}{10}$

Step 6.2.3.7

Write $1$ as a fraction with a common denominator.

$\frac{10}{10} - \frac{1}{10}$

Step 6.2.3.8

Combine the numerators over the common denominator.

$\frac{10 - 1}{10}$

Step 6.2.3.9

Subtract $1$ from $10$ .

$\frac{9}{10}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$\frac{9}{10}$

Decimal Form:

$0.9$

Step 8