Calculus Examples

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

Step 1

Remove parentheses.

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

Step 2

Split the single integral into multiple integrals.

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

Step 3

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

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

Step 4

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

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

Step 5

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

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

Step 6

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

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

Step 7

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

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

Step 8

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

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

Step 9

Apply the constant rule.

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

Step 10

Substitute and simplify.

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Step 10.1

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

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

Step 10.2

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

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

Step 10.3

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

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

Step 10.4

Simplify.

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Step 10.4.1

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

$- (\frac{81}{4} - \frac{{(- 1)}^{4}}{4}) + 3 (\frac{3^{3}}{3} - \frac{{(- 1)}^{3}}{3}) + (3) + 1$

Step 10.4.2

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

$- (\frac{81}{4} - \frac{1}{4}) + 3 (\frac{3^{3}}{3} - \frac{{(- 1)}^{3}}{3}) + (3) + 1$

Step 10.4.3

Combine the numerators over the common denominator.

$- \frac{81 - 1}{4} + 3 (\frac{3^{3}}{3} - \frac{{(- 1)}^{3}}{3}) + (3) + 1$

Step 10.4.4

Subtract $1$ from $81$ .

$- \frac{80}{4} + 3 (\frac{3^{3}}{3} - \frac{{(- 1)}^{3}}{3}) + (3) + 1$

Step 10.4.5

Cancel the common factor of $80$ and $4$ .

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Step 10.4.5.1

Factor $4$ out of $80$ .

$- \frac{4 \cdot 20}{4} + 3 (\frac{3^{3}}{3} - \frac{{(- 1)}^{3}}{3}) + (3) + 1$

Step 10.4.5.2

Cancel the common factors.

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Step 10.4.5.2.1

Factor $4$ out of $4$ .

$- \frac{4 \cdot 20}{4 (1)} + 3 (\frac{3^{3}}{3} - \frac{{(- 1)}^{3}}{3}) + (3) + 1$

Step 10.4.5.2.2

Cancel the common factor.

$- \frac{4 \cdot 20}{4 \cdot 1} + 3 (\frac{3^{3}}{3} - \frac{{(- 1)}^{3}}{3}) + (3) + 1$

Step 10.4.5.2.3

Rewrite the expression.

$- \frac{20}{1} + 3 (\frac{3^{3}}{3} - \frac{{(- 1)}^{3}}{3}) + (3) + 1$

Step 10.4.5.2.4

Divide $20$ by $1$ .

$- 1 \cdot 20 + 3 (\frac{3^{3}}{3} - \frac{{(- 1)}^{3}}{3}) + (3) + 1$

Step 10.4.6

Multiply $- 1$ by $20$ .

$- 20 + 3 (\frac{3^{3}}{3} - \frac{{(- 1)}^{3}}{3}) + (3) + 1$

Step 10.4.7

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

$- 20 + 3 (\frac{27}{3} - \frac{{(- 1)}^{3}}{3}) + (3) + 1$

Step 10.4.8

Cancel the common factor of $27$ and $3$ .

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Step 10.4.8.1

Factor $3$ out of $27$ .

$- 20 + 3 (\frac{3 \cdot 9}{3} - \frac{{(- 1)}^{3}}{3}) + (3) + 1$

Step 10.4.8.2

Cancel the common factors.

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Step 10.4.8.2.1

Factor $3$ out of $3$ .

$- 20 + 3 (\frac{3 \cdot 9}{3 (1)} - \frac{{(- 1)}^{3}}{3}) + (3) + 1$

Step 10.4.8.2.2

Cancel the common factor.

$- 20 + 3 (\frac{3 \cdot 9}{3 \cdot 1} - \frac{{(- 1)}^{3}}{3}) + (3) + 1$

Step 10.4.8.2.3

Rewrite the expression.

$- 20 + 3 (\frac{9}{1} - \frac{{(- 1)}^{3}}{3}) + (3) + 1$

Step 10.4.8.2.4

Divide $9$ by $1$ .

$- 20 + 3 (9 - \frac{{(- 1)}^{3}}{3}) + (3) + 1$

Step 10.4.9

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

$- 20 + 3 (9 - \frac{- 1}{3}) + (3) + 1$

Step 10.4.10

Move the negative in front of the fraction.

$- 20 + 3 (9 - - \frac{1}{3}) + (3) + 1$

Step 10.4.11

Multiply $- 1$ by $- 1$ .

$- 20 + 3 (9 + 1 (\frac{1}{3})) + (3) + 1$

Step 10.4.12

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

$- 20 + 3 (9 + \frac{1}{3}) + (3) + 1$

Step 10.4.13

To write $9$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$- 20 + 3 (9 \cdot \frac{3}{3} + \frac{1}{3}) + (3) + 1$

Step 10.4.14

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

$- 20 + 3 (\frac{9 \cdot 3}{3} + \frac{1}{3}) + (3) + 1$

Step 10.4.15

Combine the numerators over the common denominator.

$- 20 + 3 \frac{9 \cdot 3 + 1}{3} + (3) + 1$

Step 10.4.16

Simplify the numerator.

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Step 10.4.16.1

Multiply $9$ by $3$ .

$- 20 + 3 \frac{27 + 1}{3} + (3) + 1$

Step 10.4.16.2

Add $27$ and $1$ .

$- 20 + 3 (\frac{28}{3}) + (3) + 1$

Step 10.4.17

Combine $3$ and $\frac{28}{3}$ .

$- 20 + \frac{3 \cdot 28}{3} + (3) + 1$

Step 10.4.18

Multiply $3$ by $28$ .

$- 20 + \frac{84}{3} + (3) + 1$

Step 10.4.19

Cancel the common factor of $84$ and $3$ .

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Step 10.4.19.1

Factor $3$ out of $84$ .

$- 20 + \frac{3 \cdot 28}{3} + (3) + 1$

Step 10.4.19.2

Cancel the common factors.

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Step 10.4.19.2.1

Factor $3$ out of $3$ .

$- 20 + \frac{3 \cdot 28}{3 (1)} + (3) + 1$

Step 10.4.19.2.2

Cancel the common factor.

$- 20 + \frac{3 \cdot 28}{3 \cdot 1} + (3) + 1$

Step 10.4.19.2.3

Rewrite the expression.

$- 20 + \frac{28}{1} + (3) + 1$

Step 10.4.19.2.4

Divide $28$ by $1$ .

$- 20 + 28 + (3) + 1$

Step 10.4.20

Add $- 20$ and $28$ .

$8 + (3) + 1$

Step 10.4.21

Add $3$ and $1$ .

$8 + 4$

Step 10.4.22

Add $8$ and $4$ .

$12$

Step 11