Calculus Examples

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

Step 1

Simplify.

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Step 1.1

Apply the distributive property.

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

Step 1.2

Apply the distributive property.

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

Step 1.3

Apply the distributive property.

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

Step 1.4

Move $x^{2}$ .

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

Step 1.5

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

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

Step 1.6

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

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

Step 1.7

Add $2$ and $1$ .

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

Step 1.8

Multiply $4$ by $3$ .

$\int_{- 1}^{1} 4 x^{3} + 12 x^{2} - 1 x - 1 \cdot 3 d x$

Step 1.9

Multiply $- 1$ by $3$ .

$\int_{- 1}^{1} 4 x^{3} + 12 x^{2} - 1 x - 3 d x$

Step 2

Rewrite $- 1 x$ as $- x$ .

$\int_{- 1}^{1} 4 x^{3} + 12 x^{2} - x - 3 d x$

Step 3

Split the single integral into multiple integrals.

$\int_{- 1}^{1} 4 x^{3} d x + \int_{- 1}^{1} 12 x^{2} d x + \int_{- 1}^{1} - x d x + \int_{- 1}^{1} - 3 d x$

Step 4

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

$4 \int_{- 1}^{1} x^{3} d x + \int_{- 1}^{1} 12 x^{2} d x + \int_{- 1}^{1} - x d x + \int_{- 1}^{1} - 3 d x$

Step 5

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

$4 (\frac{1}{4} x^{4}]_{- 1}^{1}) + \int_{- 1}^{1} 12 x^{2} d x + \int_{- 1}^{1} - x d x + \int_{- 1}^{1} - 3 d x$

Step 6

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

$4 (\frac{x^{4}}{4}]_{- 1}^{1}) + \int_{- 1}^{1} 12 x^{2} d x + \int_{- 1}^{1} - x d x + \int_{- 1}^{1} - 3 d x$

Step 7

Since $12$ is constant with respect to $x$ , move $12$ out of the integral.

$4 (\frac{x^{4}}{4}]_{- 1}^{1}) + 12 \int_{- 1}^{1} x^{2} d x + \int_{- 1}^{1} - x d x + \int_{- 1}^{1} - 3 d x$

Step 8

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

$4 (\frac{x^{4}}{4}]_{- 1}^{1}) + 12 (\frac{1}{3} x^{3}]_{- 1}^{1}) + \int_{- 1}^{1} - x d x + \int_{- 1}^{1} - 3 d x$

Step 9

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

$4 (\frac{x^{4}}{4}]_{- 1}^{1}) + 12 (\frac{x^{3}}{3}]_{- 1}^{1}) + \int_{- 1}^{1} - x d x + \int_{- 1}^{1} - 3 d x$

Step 10

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

$4 (\frac{x^{4}}{4}]_{- 1}^{1}) + 12 (\frac{x^{3}}{3}]_{- 1}^{1}) - \int_{- 1}^{1} x d x + \int_{- 1}^{1} - 3 d x$

Step 11

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

$4 (\frac{x^{4}}{4}]_{- 1}^{1}) + 12 (\frac{x^{3}}{3}]_{- 1}^{1}) - (\frac{1}{2} x^{2}]_{- 1}^{1}) + \int_{- 1}^{1} - 3 d x$

Step 12

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

$4 (\frac{x^{4}}{4}]_{- 1}^{1}) + 12 (\frac{x^{3}}{3}]_{- 1}^{1}) - (\frac{x^{2}}{2}]_{- 1}^{1}) + \int_{- 1}^{1} - 3 d x$

Step 13

Apply the constant rule.

$4 (\frac{x^{4}}{4}]_{- 1}^{1}) + 12 (\frac{x^{3}}{3}]_{- 1}^{1}) - (\frac{x^{2}}{2}]_{- 1}^{1}) + - 3 x]_{- 1}^{1}$

Step 14

Substitute and simplify.

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Step 14.1

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

$4 ((\frac{1^{4}}{4}) - \frac{{(- 1)}^{4}}{4}) + 12 (\frac{x^{3}}{3}]_{- 1}^{1}) - (\frac{x^{2}}{2}]_{- 1}^{1}) + - 3 x]_{- 1}^{1}$

Step 14.2

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

$4 (\frac{1^{4}}{4} - \frac{{(- 1)}^{4}}{4}) + 12 (\frac{1^{3}}{3} - \frac{{(- 1)}^{3}}{3}) - (\frac{x^{2}}{2}]_{- 1}^{1}) + - 3 x]_{- 1}^{1}$

Step 14.3

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

$4 (\frac{1^{4}}{4} - \frac{{(- 1)}^{4}}{4}) + 12 (\frac{1^{3}}{3} - \frac{{(- 1)}^{3}}{3}) - ((\frac{1^{2}}{2}) - \frac{{(- 1)}^{2}}{2}) + - 3 x]_{- 1}^{1}$

Step 14.4

Evaluate $- 3 x$ at $1$ and at $- 1$ .

$4 (\frac{1^{4}}{4} - \frac{{(- 1)}^{4}}{4}) + 12 (\frac{1^{3}}{3} - \frac{{(- 1)}^{3}}{3}) - (\frac{1^{2}}{2} - \frac{{(- 1)}^{2}}{2}) + - 3 \cdot 1 + 3 \cdot - 1$

Step 14.5

Simplify.

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Step 14.5.1

One to any power is one.

$4 (\frac{1}{4} - \frac{{(- 1)}^{4}}{4}) + 12 (\frac{1^{3}}{3} - \frac{{(- 1)}^{3}}{3}) - (\frac{1^{2}}{2} - \frac{{(- 1)}^{2}}{2}) - 3 \cdot 1 + 3 \cdot - 1$

Step 14.5.2

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

$4 (\frac{1}{4} - \frac{1}{4}) + 12 (\frac{1^{3}}{3} - \frac{{(- 1)}^{3}}{3}) - (\frac{1^{2}}{2} - \frac{{(- 1)}^{2}}{2}) - 3 \cdot 1 + 3 \cdot - 1$

Step 14.5.3

Combine the numerators over the common denominator.

$4 \frac{1 - 1}{4} + 12 (\frac{1^{3}}{3} - \frac{{(- 1)}^{3}}{3}) - (\frac{1^{2}}{2} - \frac{{(- 1)}^{2}}{2}) - 3 \cdot 1 + 3 \cdot - 1$

Step 14.5.4

Subtract $1$ from $1$ .

$4 (\frac{0}{4}) + 12 (\frac{1^{3}}{3} - \frac{{(- 1)}^{3}}{3}) - (\frac{1^{2}}{2} - \frac{{(- 1)}^{2}}{2}) - 3 \cdot 1 + 3 \cdot - 1$

Step 14.5.5

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

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Step 14.5.5.1

Factor $4$ out of $0$ .

$4 \frac{4 (0)}{4} + 12 (\frac{1^{3}}{3} - \frac{{(- 1)}^{3}}{3}) - (\frac{1^{2}}{2} - \frac{{(- 1)}^{2}}{2}) - 3 \cdot 1 + 3 \cdot - 1$

Step 14.5.5.2

Cancel the common factors.

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Step 14.5.5.2.1

Factor $4$ out of $4$ .

$4 \frac{4 \cdot 0}{4 \cdot 1} + 12 (\frac{1^{3}}{3} - \frac{{(- 1)}^{3}}{3}) - (\frac{1^{2}}{2} - \frac{{(- 1)}^{2}}{2}) - 3 \cdot 1 + 3 \cdot - 1$

Step 14.5.5.2.2

Cancel the common factor.

$4 \frac{4 \cdot 0}{4 \cdot 1} + 12 (\frac{1^{3}}{3} - \frac{{(- 1)}^{3}}{3}) - (\frac{1^{2}}{2} - \frac{{(- 1)}^{2}}{2}) - 3 \cdot 1 + 3 \cdot - 1$

Step 14.5.5.2.3

Rewrite the expression.

$4 (\frac{0}{1}) + 12 (\frac{1^{3}}{3} - \frac{{(- 1)}^{3}}{3}) - (\frac{1^{2}}{2} - \frac{{(- 1)}^{2}}{2}) - 3 \cdot 1 + 3 \cdot - 1$

Step 14.5.5.2.4

Divide $0$ by $1$ .

$4 \cdot 0 + 12 (\frac{1^{3}}{3} - \frac{{(- 1)}^{3}}{3}) - (\frac{1^{2}}{2} - \frac{{(- 1)}^{2}}{2}) - 3 \cdot 1 + 3 \cdot - 1$

Step 14.5.6

Multiply $4$ by $0$ .

$0 + 12 (\frac{1^{3}}{3} - \frac{{(- 1)}^{3}}{3}) - (\frac{1^{2}}{2} - \frac{{(- 1)}^{2}}{2}) - 3 \cdot 1 + 3 \cdot - 1$

Step 14.5.7

One to any power is one.

$0 + 12 (\frac{1}{3} - \frac{{(- 1)}^{3}}{3}) - (\frac{1^{2}}{2} - \frac{{(- 1)}^{2}}{2}) - 3 \cdot 1 + 3 \cdot - 1$

Step 14.5.8

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

$0 + 12 (\frac{1}{3} - \frac{- 1}{3}) - (\frac{1^{2}}{2} - \frac{{(- 1)}^{2}}{2}) - 3 \cdot 1 + 3 \cdot - 1$

Step 14.5.9

Move the negative in front of the fraction.

$0 + 12 (\frac{1}{3} - - \frac{1}{3}) - (\frac{1^{2}}{2} - \frac{{(- 1)}^{2}}{2}) - 3 \cdot 1 + 3 \cdot - 1$

Step 14.5.10

Multiply $- 1$ by $- 1$ .

$0 + 12 (\frac{1}{3} + 1 (\frac{1}{3})) - (\frac{1^{2}}{2} - \frac{{(- 1)}^{2}}{2}) - 3 \cdot 1 + 3 \cdot - 1$

Step 14.5.11

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

$0 + 12 (\frac{1}{3} + \frac{1}{3}) - (\frac{1^{2}}{2} - \frac{{(- 1)}^{2}}{2}) - 3 \cdot 1 + 3 \cdot - 1$

Step 14.5.12

Combine the numerators over the common denominator.

$0 + 12 \frac{1 + 1}{3} - (\frac{1^{2}}{2} - \frac{{(- 1)}^{2}}{2}) - 3 \cdot 1 + 3 \cdot - 1$

Step 14.5.13

Add $1$ and $1$ .

$0 + 12 (\frac{2}{3}) - (\frac{1^{2}}{2} - \frac{{(- 1)}^{2}}{2}) - 3 \cdot 1 + 3 \cdot - 1$

Step 14.5.14

Combine $12$ and $\frac{2}{3}$ .

$0 + \frac{12 \cdot 2}{3} - (\frac{1^{2}}{2} - \frac{{(- 1)}^{2}}{2}) - 3 \cdot 1 + 3 \cdot - 1$

Step 14.5.15

Multiply $12$ by $2$ .

$0 + \frac{24}{3} - (\frac{1^{2}}{2} - \frac{{(- 1)}^{2}}{2}) - 3 \cdot 1 + 3 \cdot - 1$

Step 14.5.16

Cancel the common factor of $24$ and $3$ .

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Step 14.5.16.1

Factor $3$ out of $24$ .

$0 + \frac{3 \cdot 8}{3} - (\frac{1^{2}}{2} - \frac{{(- 1)}^{2}}{2}) - 3 \cdot 1 + 3 \cdot - 1$

Step 14.5.16.2

Cancel the common factors.

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Step 14.5.16.2.1

Factor $3$ out of $3$ .

$0 + \frac{3 \cdot 8}{3 (1)} - (\frac{1^{2}}{2} - \frac{{(- 1)}^{2}}{2}) - 3 \cdot 1 + 3 \cdot - 1$

Step 14.5.16.2.2

Cancel the common factor.

$0 + \frac{3 \cdot 8}{3 \cdot 1} - (\frac{1^{2}}{2} - \frac{{(- 1)}^{2}}{2}) - 3 \cdot 1 + 3 \cdot - 1$

Step 14.5.16.2.3

Rewrite the expression.

$0 + \frac{8}{1} - (\frac{1^{2}}{2} - \frac{{(- 1)}^{2}}{2}) - 3 \cdot 1 + 3 \cdot - 1$

Step 14.5.16.2.4

Divide $8$ by $1$ .

$0 + 8 - (\frac{1^{2}}{2} - \frac{{(- 1)}^{2}}{2}) - 3 \cdot 1 + 3 \cdot - 1$

Step 14.5.17

Add $0$ and $8$ .

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

Step 14.5.18

One to any power is one.

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

Step 14.5.19

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

$8 - (\frac{1}{2} - \frac{1}{2}) - 3 \cdot 1 + 3 \cdot - 1$

Step 14.5.20

Combine the numerators over the common denominator.

$8 - \frac{1 - 1}{2} - 3 \cdot 1 + 3 \cdot - 1$

Step 14.5.21

Subtract $1$ from $1$ .

$8 - \frac{0}{2} - 3 \cdot 1 + 3 \cdot - 1$

Step 14.5.22

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

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Step 14.5.22.1

Factor $2$ out of $0$ .

$8 - \frac{2 (0)}{2} - 3 \cdot 1 + 3 \cdot - 1$

Step 14.5.22.2

Cancel the common factors.

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Step 14.5.22.2.1

Factor $2$ out of $2$ .

$8 - \frac{2 \cdot 0}{2 \cdot 1} - 3 \cdot 1 + 3 \cdot - 1$

Step 14.5.22.2.2

Cancel the common factor.

$8 - \frac{2 \cdot 0}{2 \cdot 1} - 3 \cdot 1 + 3 \cdot - 1$

Step 14.5.22.2.3

Rewrite the expression.

$8 - \frac{0}{1} - 3 \cdot 1 + 3 \cdot - 1$

Step 14.5.22.2.4

Divide $0$ by $1$ .

$8 - 0 - 3 \cdot 1 + 3 \cdot - 1$

Step 14.5.23

Multiply $- 1$ by $0$ .

$8 + 0 - 3 \cdot 1 + 3 \cdot - 1$

Step 14.5.24

Add $8$ and $0$ .

$8 - 3 \cdot 1 + 3 \cdot - 1$

Step 14.5.25

Multiply $- 3$ by $1$ .

$8 - 3 + 3 \cdot - 1$

Step 14.5.26

Multiply $3$ by $- 1$ .

$8 - 3 - 3$

Step 14.5.27

Subtract $3$ from $- 3$ .

$8 - 6$

Step 14.5.28

Subtract $6$ from $8$ .

$2$

Step 15