Calculus Examples

$\int_{- 1}^{1} \frac{9 x^{3} - 8 x^{2}}{4 x} d x$

Step 1

Since $\frac{1}{4}$ is constant with respect to $x$ , move $\frac{1}{4}$ out of the integral.

$\frac{1}{4} \int_{- 1}^{1} \frac{9 x^{3} - 8 x^{2}}{x} d x$

Step 2

Simplify.

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Step 2.1

Factor $x^{2}$ out of $9 x^{3} - 8 x^{2}$ .

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Step 2.1.1

Factor $x^{2}$ out of $9 x^{3}$ .

$\frac{1}{4} \int_{- 1}^{1} \frac{x^{2} (9 x) - 8 x^{2}}{x} d x$

Step 2.1.2

Factor $x^{2}$ out of $- 8 x^{2}$ .

$\frac{1}{4} \int_{- 1}^{1} \frac{x^{2} (9 x) + x^{2} \cdot - 8}{x} d x$

Step 2.1.3

Factor $x^{2}$ out of $x^{2} (9 x) + x^{2} \cdot - 8$ .

$\frac{1}{4} \int_{- 1}^{1} \frac{x^{2} (9 x - 8)}{x} d x$

Step 2.2

Cancel the common factor of $x^{2}$ and $x$ .

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Step 2.2.1

Factor $x$ out of $x^{2} (9 x - 8)$ .

$\frac{1}{4} \int_{- 1}^{1} \frac{x (x (9 x - 8))}{x} d x$

Step 2.2.2

Cancel the common factors.

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Step 2.2.2.1

Raise $x$ to the power of $1$ .

$\frac{1}{4} \int_{- 1}^{1} \frac{x (x (9 x - 8))}{x^{1}} d x$

Step 2.2.2.2

Factor $x$ out of $x^{1}$ .

$\frac{1}{4} \int_{- 1}^{1} \frac{x (x (9 x - 8))}{x \cdot 1} d x$

Step 2.2.2.3

Cancel the common factor.

$\frac{1}{4} \int_{- 1}^{1} \frac{x (x (9 x - 8))}{x \cdot 1} d x$

Step 2.2.2.4

Rewrite the expression.

$\frac{1}{4} \int_{- 1}^{1} \frac{x (9 x - 8)}{1} d x$

Step 2.2.2.5

Divide $x (9 x - 8)$ by $1$ .

$\frac{1}{4} \int_{- 1}^{1} x (9 x - 8) d x$

Step 3

Multiply $x (9 x - 8)$ .

$\frac{1}{4} \int_{- 1}^{1} x (9 x) + x \cdot - 8 d x$

Step 4

Simplify.

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Step 4.1

Raise $x$ to the power of $1$ .

$\frac{1}{4} \int_{- 1}^{1} 9 (x^{1} x) + x \cdot - 8 d x$

Step 4.2

Raise $x$ to the power of $1$ .

$\frac{1}{4} \int_{- 1}^{1} 9 (x^{1} x^{1}) + x \cdot - 8 d x$

Step 4.3

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{1}{4} \int_{- 1}^{1} 9 x^{1 + 1} + x \cdot - 8 d x$

Step 4.4

Add $1$ and $1$ .

$\frac{1}{4} \int_{- 1}^{1} 9 x^{2} + x \cdot - 8 d x$

Step 4.5

Move $- 8$ to the left of $x$ .

$\frac{1}{4} \int_{- 1}^{1} 9 x^{2} - 8 x d x$

Step 5

Split the single integral into multiple integrals.

$\frac{1}{4} (\int_{- 1}^{1} 9 x^{2} d x + \int_{- 1}^{1} - 8 x d x)$

Step 6

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

$\frac{1}{4} (9 \int_{- 1}^{1} x^{2} d x + \int_{- 1}^{1} - 8 x d x)$

Step 7

By the Power Rule, the integral of $x^{2}$ with respect to $x$ is $\frac{1}{3} x^{3}$ .

$\frac{1}{4} (9 (\frac{1}{3} x^{3}]_{- 1}^{1}) + \int_{- 1}^{1} - 8 x d x)$

Step 8

Combine $\frac{1}{3}$ and $x^{3}$ .

$\frac{1}{4} (9 (\frac{x^{3}}{3}]_{- 1}^{1}) + \int_{- 1}^{1} - 8 x d x)$

Step 9

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

$\frac{1}{4} (9 (\frac{x^{3}}{3}]_{- 1}^{1}) - 8 \int_{- 1}^{1} x d x)$

Step 10

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

$\frac{1}{4} (9 (\frac{x^{3}}{3}]_{- 1}^{1}) - 8 (\frac{1}{2} x^{2}]_{- 1}^{1}))$

Step 11

Simplify the answer.

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Step 11.1

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

$\frac{1}{4} (9 (\frac{x^{3}}{3}]_{- 1}^{1}) - 8 (\frac{x^{2}}{2}]_{- 1}^{1}))$

Step 11.2

Substitute and simplify.

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Step 11.2.1

Evaluate $\frac{x^{3}}{3}$ at $1$ and at $- 1$ .

$\frac{1}{4} (9 ((\frac{1^{3}}{3}) - \frac{{(- 1)}^{3}}{3}) - 8 (\frac{x^{2}}{2}]_{- 1}^{1}))$

Step 11.2.2

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

$\frac{1}{4} (9 (\frac{1^{3}}{3} - \frac{{(- 1)}^{3}}{3}) - 8 (\frac{1^{2}}{2} - \frac{{(- 1)}^{2}}{2}))$

Step 11.2.3

Simplify.

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Step 11.2.3.1

One to any power is one.

$\frac{1}{4} (9 (\frac{1}{3} - \frac{{(- 1)}^{3}}{3}) - 8 (\frac{1^{2}}{2} - \frac{{(- 1)}^{2}}{2}))$

Step 11.2.3.2

Raise $- 1$ to the power of $3$ .

$\frac{1}{4} (9 (\frac{1}{3} - \frac{- 1}{3}) - 8 (\frac{1^{2}}{2} - \frac{{(- 1)}^{2}}{2}))$

Step 11.2.3.3

Move the negative in front of the fraction.

$\frac{1}{4} (9 (\frac{1}{3} - - \frac{1}{3}) - 8 (\frac{1^{2}}{2} - \frac{{(- 1)}^{2}}{2}))$

Step 11.2.3.4

Multiply $- 1$ by $- 1$ .

$\frac{1}{4} (9 (\frac{1}{3} + 1 (\frac{1}{3})) - 8 (\frac{1^{2}}{2} - \frac{{(- 1)}^{2}}{2}))$

Step 11.2.3.5

Multiply $\frac{1}{3}$ by $1$ .

$\frac{1}{4} (9 (\frac{1}{3} + \frac{1}{3}) - 8 (\frac{1^{2}}{2} - \frac{{(- 1)}^{2}}{2}))$

Step 11.2.3.6

Combine the numerators over the common denominator.

$\frac{1}{4} (9 \frac{1 + 1}{3} - 8 (\frac{1^{2}}{2} - \frac{{(- 1)}^{2}}{2}))$

Step 11.2.3.7

Add $1$ and $1$ .

$\frac{1}{4} (9 (\frac{2}{3}) - 8 (\frac{1^{2}}{2} - \frac{{(- 1)}^{2}}{2}))$

Step 11.2.3.8

Combine $9$ and $\frac{2}{3}$ .

$\frac{1}{4} (\frac{9 \cdot 2}{3} - 8 (\frac{1^{2}}{2} - \frac{{(- 1)}^{2}}{2}))$

Step 11.2.3.9

Multiply $9$ by $2$ .

$\frac{1}{4} (\frac{18}{3} - 8 (\frac{1^{2}}{2} - \frac{{(- 1)}^{2}}{2}))$

Step 11.2.3.10

Cancel the common factor of $18$ and $3$ .

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Step 11.2.3.10.1

Factor $3$ out of $18$ .

$\frac{1}{4} (\frac{3 \cdot 6}{3} - 8 (\frac{1^{2}}{2} - \frac{{(- 1)}^{2}}{2}))$

Step 11.2.3.10.2

Cancel the common factors.

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Step 11.2.3.10.2.1

Factor $3$ out of $3$ .

$\frac{1}{4} (\frac{3 \cdot 6}{3 (1)} - 8 (\frac{1^{2}}{2} - \frac{{(- 1)}^{2}}{2}))$

Step 11.2.3.10.2.2

Cancel the common factor.

$\frac{1}{4} (\frac{3 \cdot 6}{3 \cdot 1} - 8 (\frac{1^{2}}{2} - \frac{{(- 1)}^{2}}{2}))$

Step 11.2.3.10.2.3

Rewrite the expression.

$\frac{1}{4} (\frac{6}{1} - 8 (\frac{1^{2}}{2} - \frac{{(- 1)}^{2}}{2}))$

Step 11.2.3.10.2.4

Divide $6$ by $1$ .

$\frac{1}{4} (6 - 8 (\frac{1^{2}}{2} - \frac{{(- 1)}^{2}}{2}))$

Step 11.2.3.11

One to any power is one.

$\frac{1}{4} (6 - 8 (\frac{1}{2} - \frac{{(- 1)}^{2}}{2}))$

Step 11.2.3.12

Raise $- 1$ to the power of $2$ .

$\frac{1}{4} (6 - 8 (\frac{1}{2} - \frac{1}{2}))$

Step 11.2.3.13

Combine the numerators over the common denominator.

$\frac{1}{4} (6 - 8 \frac{1 - 1}{2})$

Step 11.2.3.14

Subtract $1$ from $1$ .

$\frac{1}{4} (6 - 8 (\frac{0}{2}))$

Step 11.2.3.15

Cancel the common factor of $0$ and $2$ .

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Step 11.2.3.15.1

Factor $2$ out of $0$ .

$\frac{1}{4} (6 - 8 \frac{2 (0)}{2})$

Step 11.2.3.15.2

Cancel the common factors.

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Step 11.2.3.15.2.1

Factor $2$ out of $2$ .

$\frac{1}{4} (6 - 8 \frac{2 \cdot 0}{2 \cdot 1})$

Step 11.2.3.15.2.2

Cancel the common factor.

$\frac{1}{4} (6 - 8 \frac{2 \cdot 0}{2 \cdot 1})$

Step 11.2.3.15.2.3

Rewrite the expression.

$\frac{1}{4} (6 - 8 (\frac{0}{1}))$

Step 11.2.3.15.2.4

Divide $0$ by $1$ .

$\frac{1}{4} (6 - 8 \cdot 0)$

Step 11.2.3.16

Multiply $- 8$ by $0$ .

$\frac{1}{4} (6 + 0)$

Step 11.2.3.17

Add $6$ and $0$ .

$\frac{1}{4} \cdot 6$

Step 11.2.3.18

Combine $\frac{1}{4}$ and $6$ .

$\frac{6}{4}$

Step 11.2.3.19

Cancel the common factor of $6$ and $4$ .

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Step 11.2.3.19.1

Factor $2$ out of $6$ .

$\frac{2 (3)}{4}$

Step 11.2.3.19.2

Cancel the common factors.

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Step 11.2.3.19.2.1

Factor $2$ out of $4$ .

$\frac{2 \cdot 3}{2 \cdot 2}$

Step 11.2.3.19.2.2

Cancel the common factor.

$\frac{2 \cdot 3}{2 \cdot 2}$

Step 11.2.3.19.2.3

Rewrite the expression.

$\frac{3}{2}$

Step 12

The result can be shown in multiple forms.

Exact Form:

$\frac{3}{2}$

Decimal Form:

$1.5$

Mixed Number Form:

$1 \frac{1}{2}$

Step 13