Calculus Examples

$\int_{- 4}^{- 1} x^{2} (3 - 4 x) d x$

Step 1

Multiply $x^{2} (3 - 4 x)$ .

$\int_{- 4}^{- 1} x^{2} \cdot 3 + x^{2} (- 4 x) d x$

Step 2

Simplify.

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Step 2.1

Move $3$ to the left of $x^{2}$ .

$\int_{- 4}^{- 1} 3 \cdot x^{2} + x^{2} (- 4 x) d x$

Step 2.2

Multiply $x^{2}$ by $x$ by adding the exponents.

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Step 2.2.1

Move $x$ .

$\int_{- 4}^{- 1} 3 x^{2} + x \cdot x^{2} \cdot - 4 d x$

Step 2.2.2

Multiply $x$ by $x^{2}$ .

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Step 2.2.2.1

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

$\int_{- 4}^{- 1} 3 x^{2} + x^{1} x^{2} \cdot - 4 d x$

Step 2.2.2.2

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

$\int_{- 4}^{- 1} 3 x^{2} + x^{1 + 2} \cdot - 4 d x$

Step 2.2.3

Add $1$ and $2$ .

$\int_{- 4}^{- 1} 3 x^{2} + x^{3} \cdot - 4 d x$

Step 2.3

Move $- 4$ to the left of $x^{3}$ .

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

Step 3

Split the single integral into multiple integrals.

$\int_{- 4}^{- 1} 3 x^{2} d x + \int_{- 4}^{- 1} - 4 x^{3} d x$

Step 4

Since $3$ is constant with respect to $x$ , move $3$ out of the integral.

$3 \int_{- 4}^{- 1} x^{2} d x + \int_{- 4}^{- 1} - 4 x^{3} d x$

Step 5

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

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

Step 6

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

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

Step 7

Since $- 4$ is constant with respect to $x$ , move $- 4$ out of the integral.

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

Step 8

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

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

Step 9

Simplify the answer.

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Step 9.1

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

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

Step 9.2

Substitute and simplify.

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Step 9.2.1

Evaluate $\frac{x^{3}}{3}$ at $- 1$ and at $- 4$ .

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

Step 9.2.2

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

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

Step 9.2.3

Simplify.

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Step 9.2.3.1

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

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

Step 9.2.3.2

Move the negative in front of the fraction.

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

Step 9.2.3.3

Raise $- 4$ to the power of $3$ .

$3 (- \frac{1}{3} - \frac{- 64}{3}) - 4 (\frac{{(- 1)}^{4}}{4} - \frac{{(- 4)}^{4}}{4})$

Step 9.2.3.4

Move the negative in front of the fraction.

$3 (- \frac{1}{3} - - \frac{64}{3}) - 4 (\frac{{(- 1)}^{4}}{4} - \frac{{(- 4)}^{4}}{4})$

Step 9.2.3.5

Multiply $- 1$ by $- 1$ .

$3 (- \frac{1}{3} + 1 (\frac{64}{3})) - 4 (\frac{{(- 1)}^{4}}{4} - \frac{{(- 4)}^{4}}{4})$

Step 9.2.3.6

Multiply $\frac{64}{3}$ by $1$ .

$3 (- \frac{1}{3} + \frac{64}{3}) - 4 (\frac{{(- 1)}^{4}}{4} - \frac{{(- 4)}^{4}}{4})$

Step 9.2.3.7

Combine the numerators over the common denominator.

$3 \frac{- 1 + 64}{3} - 4 (\frac{{(- 1)}^{4}}{4} - \frac{{(- 4)}^{4}}{4})$

Step 9.2.3.8

Add $- 1$ and $64$ .

$3 (\frac{63}{3}) - 4 (\frac{{(- 1)}^{4}}{4} - \frac{{(- 4)}^{4}}{4})$

Step 9.2.3.9

Cancel the common factor of $63$ and $3$ .

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Step 9.2.3.9.1

Factor $3$ out of $63$ .

$3 \frac{3 \cdot 21}{3} - 4 (\frac{{(- 1)}^{4}}{4} - \frac{{(- 4)}^{4}}{4})$

Step 9.2.3.9.2

Cancel the common factors.

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Step 9.2.3.9.2.1

Factor $3$ out of $3$ .

$3 \frac{3 \cdot 21}{3 (1)} - 4 (\frac{{(- 1)}^{4}}{4} - \frac{{(- 4)}^{4}}{4})$

Step 9.2.3.9.2.2

Cancel the common factor.

$3 \frac{3 \cdot 21}{3 \cdot 1} - 4 (\frac{{(- 1)}^{4}}{4} - \frac{{(- 4)}^{4}}{4})$

Step 9.2.3.9.2.3

Rewrite the expression.

$3 (\frac{21}{1}) - 4 (\frac{{(- 1)}^{4}}{4} - \frac{{(- 4)}^{4}}{4})$

Step 9.2.3.9.2.4

Divide $21$ by $1$ .

$3 \cdot 21 - 4 (\frac{{(- 1)}^{4}}{4} - \frac{{(- 4)}^{4}}{4})$

Step 9.2.3.10

Multiply $3$ by $21$ .

$63 - 4 (\frac{{(- 1)}^{4}}{4} - \frac{{(- 4)}^{4}}{4})$

Step 9.2.3.11

Raise $- 1$ to the power of $4$ .

$63 - 4 (\frac{1}{4} - \frac{{(- 4)}^{4}}{4})$

Step 9.2.3.12

Raise $- 4$ to the power of $4$ .

$63 - 4 (\frac{1}{4} - \frac{256}{4})$

Step 9.2.3.13

Cancel the common factor of $256$ and $4$ .

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Step 9.2.3.13.1

Factor $4$ out of $256$ .

$63 - 4 (\frac{1}{4} - \frac{4 \cdot 64}{4})$

Step 9.2.3.13.2

Cancel the common factors.

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Step 9.2.3.13.2.1

Factor $4$ out of $4$ .

$63 - 4 (\frac{1}{4} - \frac{4 \cdot 64}{4 (1)})$

Step 9.2.3.13.2.2

Cancel the common factor.

$63 - 4 (\frac{1}{4} - \frac{4 \cdot 64}{4 \cdot 1})$

Step 9.2.3.13.2.3

Rewrite the expression.

$63 - 4 (\frac{1}{4} - \frac{64}{1})$

Step 9.2.3.13.2.4

Divide $64$ by $1$ .

$63 - 4 (\frac{1}{4} - 1 \cdot 64)$

Step 9.2.3.14

Multiply $- 1$ by $64$ .

$63 - 4 (\frac{1}{4} - 64)$

Step 9.2.3.15

To write $- 64$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$63 - 4 (\frac{1}{4} - 64 \cdot \frac{4}{4})$

Step 9.2.3.16

Combine $- 64$ and $\frac{4}{4}$ .

$63 - 4 (\frac{1}{4} + \frac{- 64 \cdot 4}{4})$

Step 9.2.3.17

Combine the numerators over the common denominator.

$63 - 4 \frac{1 - 64 \cdot 4}{4}$

Step 9.2.3.18

Simplify the numerator.

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Step 9.2.3.18.1

Multiply $- 64$ by $4$ .

$63 - 4 \frac{1 - 256}{4}$

Step 9.2.3.18.2

Subtract $256$ from $1$ .

$63 - 4 (\frac{- 255}{4})$

Step 9.2.3.19

Move the negative in front of the fraction.

$63 - 4 (- \frac{255}{4})$

Step 9.2.3.20

Multiply $- 1$ by $- 4$ .

$63 + 4 (\frac{255}{4})$

Step 9.2.3.21

Combine $4$ and $\frac{255}{4}$ .

$63 + \frac{4 \cdot 255}{4}$

Step 9.2.3.22

Multiply $4$ by $255$ .

$63 + \frac{1020}{4}$

Step 9.2.3.23

Cancel the common factor of $1020$ and $4$ .

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Step 9.2.3.23.1

Factor $4$ out of $1020$ .

$63 + \frac{4 \cdot 255}{4}$

Step 9.2.3.23.2

Cancel the common factors.

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Step 9.2.3.23.2.1

Factor $4$ out of $4$ .

$63 + \frac{4 \cdot 255}{4 (1)}$

Step 9.2.3.23.2.2

Cancel the common factor.

$63 + \frac{4 \cdot 255}{4 \cdot 1}$

Step 9.2.3.23.2.3

Rewrite the expression.

$63 + \frac{255}{1}$

Step 9.2.3.23.2.4

Divide $255$ by $1$ .

$63 + 255$

Step 9.2.3.24

Add $63$ and $255$ .

$318$

Step 10