Calculus Examples

$\int_{0}^{1} (1 - 8 x^{3} + 16 x^{7}) d x$

Step 1

Remove parentheses.

$\int_{0}^{1} 1 - 8 x^{3} + 16 x^{7} d x$

Step 2

Split the single integral into multiple integrals.

$\int_{0}^{1} d x + \int_{0}^{1} - 8 x^{3} d x + \int_{0}^{1} 16 x^{7} d x$

Step 3

Apply the constant rule.

$x]_{0}^{1} + \int_{0}^{1} - 8 x^{3} d x + \int_{0}^{1} 16 x^{7} d x$

Step 4

Since $- 8$ is constant with respect to $x$ , move $- 8$ out of the integral.

$x]_{0}^{1} - 8 \int_{0}^{1} x^{3} d x + \int_{0}^{1} 16 x^{7} d x$

Step 5

By the Power Rule, the integral of $x^{3}$ with respect to $x$ is $\frac{1}{4} x^{4}$ .

$x]_{0}^{1} - 8 (\frac{1}{4} x^{4}]_{0}^{1}) + \int_{0}^{1} 16 x^{7} d x$

Step 6

Combine $\frac{1}{4}$ and $x^{4}$ .

$x]_{0}^{1} - 8 (\frac{x^{4}}{4}]_{0}^{1}) + \int_{0}^{1} 16 x^{7} d x$

Step 7

Since $16$ is constant with respect to $x$ , move $16$ out of the integral.

$x]_{0}^{1} - 8 (\frac{x^{4}}{4}]_{0}^{1}) + 16 \int_{0}^{1} x^{7} d x$

Step 8

By the Power Rule, the integral of $x^{7}$ with respect to $x$ is $\frac{1}{8} x^{8}$ .

$x]_{0}^{1} - 8 (\frac{x^{4}}{4}]_{0}^{1}) + 16 (\frac{1}{8} x^{8}]_{0}^{1})$

Step 9

Simplify the answer.

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Step 9.1

Combine $\frac{1}{8}$ and $x^{8}$ .

$x]_{0}^{1} - 8 (\frac{x^{4}}{4}]_{0}^{1}) + 16 (\frac{x^{8}}{8}]_{0}^{1})$

Step 9.2

Substitute and simplify.

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Step 9.2.1

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

$(1) + 0 - 8 (\frac{x^{4}}{4}]_{0}^{1}) + 16 (\frac{x^{8}}{8}]_{0}^{1})$

Step 9.2.2

Evaluate $\frac{x^{4}}{4}$ at $1$ and at $0$ .

$1 + 0 - 8 (\frac{1^{4}}{4} - \frac{0^{4}}{4}) + 16 (\frac{x^{8}}{8}]_{0}^{1})$

Step 9.2.3

Evaluate $\frac{x^{8}}{8}$ at $1$ and at $0$ .

$1 + 0 - 8 (\frac{1^{4}}{4} - \frac{0^{4}}{4}) + 16 (\frac{1^{8}}{8} - \frac{0^{8}}{8})$

Step 9.2.4

Simplify.

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Step 9.2.4.1

Add $1$ and $0$ .

$1 - 8 (\frac{1^{4}}{4} - \frac{0^{4}}{4}) + 16 (\frac{1^{8}}{8} - \frac{0^{8}}{8})$

Step 9.2.4.2

One to any power is one.

$1 - 8 (\frac{1}{4} - \frac{0^{4}}{4}) + 16 (\frac{1^{8}}{8} - \frac{0^{8}}{8})$

Step 9.2.4.3

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

$1 - 8 (\frac{1}{4} - \frac{0}{4}) + 16 (\frac{1^{8}}{8} - \frac{0^{8}}{8})$

Step 9.2.4.4

Cancel the common factor of $0$ and $4$ .

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Step 9.2.4.4.1

Factor $4$ out of $0$ .

$1 - 8 (\frac{1}{4} - \frac{4 (0)}{4}) + 16 (\frac{1^{8}}{8} - \frac{0^{8}}{8})$

Step 9.2.4.4.2

Cancel the common factors.

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Step 9.2.4.4.2.1

Factor $4$ out of $4$ .

$1 - 8 (\frac{1}{4} - \frac{4 \cdot 0}{4 \cdot 1}) + 16 (\frac{1^{8}}{8} - \frac{0^{8}}{8})$

Step 9.2.4.4.2.2

Cancel the common factor.

$1 - 8 (\frac{1}{4} - \frac{4 \cdot 0}{4 \cdot 1}) + 16 (\frac{1^{8}}{8} - \frac{0^{8}}{8})$

Step 9.2.4.4.2.3

Rewrite the expression.

$1 - 8 (\frac{1}{4} - \frac{0}{1}) + 16 (\frac{1^{8}}{8} - \frac{0^{8}}{8})$

Step 9.2.4.4.2.4

Divide $0$ by $1$ .

$1 - 8 (\frac{1}{4} - 0) + 16 (\frac{1^{8}}{8} - \frac{0^{8}}{8})$

Step 9.2.4.5

Multiply $- 1$ by $0$ .

$1 - 8 (\frac{1}{4} + 0) + 16 (\frac{1^{8}}{8} - \frac{0^{8}}{8})$

Step 9.2.4.6

Add $\frac{1}{4}$ and $0$ .

$1 - 8 (\frac{1}{4}) + 16 (\frac{1^{8}}{8} - \frac{0^{8}}{8})$

Step 9.2.4.7

Combine $- 8$ and $\frac{1}{4}$ .

$1 + \frac{- 8}{4} + 16 (\frac{1^{8}}{8} - \frac{0^{8}}{8})$

Step 9.2.4.8

Cancel the common factor of $- 8$ and $4$ .

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Step 9.2.4.8.1

Factor $4$ out of $- 8$ .

$1 + \frac{4 \cdot - 2}{4} + 16 (\frac{1^{8}}{8} - \frac{0^{8}}{8})$

Step 9.2.4.8.2

Cancel the common factors.

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Step 9.2.4.8.2.1

Factor $4$ out of $4$ .

$1 + \frac{4 \cdot - 2}{4 (1)} + 16 (\frac{1^{8}}{8} - \frac{0^{8}}{8})$

Step 9.2.4.8.2.2

Cancel the common factor.

$1 + \frac{4 \cdot - 2}{4 \cdot 1} + 16 (\frac{1^{8}}{8} - \frac{0^{8}}{8})$

Step 9.2.4.8.2.3

Rewrite the expression.

$1 + \frac{- 2}{1} + 16 (\frac{1^{8}}{8} - \frac{0^{8}}{8})$

Step 9.2.4.8.2.4

Divide $- 2$ by $1$ .

$1 - 2 + 16 (\frac{1^{8}}{8} - \frac{0^{8}}{8})$

Step 9.2.4.9

Subtract $2$ from $1$ .

$- 1 + 16 (\frac{1^{8}}{8} - \frac{0^{8}}{8})$

Step 9.2.4.10

One to any power is one.

$- 1 + 16 (\frac{1}{8} - \frac{0^{8}}{8})$

Step 9.2.4.11

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

$- 1 + 16 (\frac{1}{8} - \frac{0}{8})$

Step 9.2.4.12

Cancel the common factor of $0$ and $8$ .

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Step 9.2.4.12.1

Factor $8$ out of $0$ .

$- 1 + 16 (\frac{1}{8} - \frac{8 (0)}{8})$

Step 9.2.4.12.2

Cancel the common factors.

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Step 9.2.4.12.2.1

Factor $8$ out of $8$ .

$- 1 + 16 (\frac{1}{8} - \frac{8 \cdot 0}{8 \cdot 1})$

Step 9.2.4.12.2.2

Cancel the common factor.

$- 1 + 16 (\frac{1}{8} - \frac{8 \cdot 0}{8 \cdot 1})$

Step 9.2.4.12.2.3

Rewrite the expression.

$- 1 + 16 (\frac{1}{8} - \frac{0}{1})$

Step 9.2.4.12.2.4

Divide $0$ by $1$ .

$- 1 + 16 (\frac{1}{8} - 0)$

Step 9.2.4.13

Multiply $- 1$ by $0$ .

$- 1 + 16 (\frac{1}{8} + 0)$

Step 9.2.4.14

Add $\frac{1}{8}$ and $0$ .

$- 1 + 16 (\frac{1}{8})$

Step 9.2.4.15

Combine $16$ and $\frac{1}{8}$ .

$- 1 + \frac{16}{8}$

Step 9.2.4.16

Cancel the common factor of $16$ and $8$ .

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Step 9.2.4.16.1

Factor $8$ out of $16$ .

$- 1 + \frac{8 \cdot 2}{8}$

Step 9.2.4.16.2

Cancel the common factors.

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Step 9.2.4.16.2.1

Factor $8$ out of $8$ .

$- 1 + \frac{8 \cdot 2}{8 (1)}$

Step 9.2.4.16.2.2

Cancel the common factor.

$- 1 + \frac{8 \cdot 2}{8 \cdot 1}$

Step 9.2.4.16.2.3

Rewrite the expression.

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

Step 9.2.4.16.2.4

Divide $2$ by $1$ .

$- 1 + 2$

Step 9.2.4.17

Add $- 1$ and $2$ .

$1$

Step 10