Calculus Examples

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

Step 1

Remove parentheses.

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

Step 2

Split the single integral into multiple integrals.

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

Step 3

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

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

Step 4

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

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

Step 5

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

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

Step 6

Since $- 2$ is constant with respect to $x$ , move $- 2$ out of the integral.

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

Step 7

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

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

Step 8

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

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

Step 9

Apply the constant rule.

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

Step 10

Substitute and simplify.

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Step 10.1

Evaluate $\frac{x^{3}}{3}$ at $1$ and at $- 2$ .

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

Step 10.2

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

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

Step 10.3

Evaluate $3 x$ at $1$ and at $- 2$ .

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

Step 10.4

Simplify.

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Step 10.4.1

One to any power is one.

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

Step 10.4.2

Raise $- 2$ to the power of $3$ .

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

Step 10.4.3

Move the negative in front of the fraction.

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

Step 10.4.4

Multiply $- 1$ by $- 1$ .

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

Step 10.4.5

Multiply $\frac{8}{3}$ by $1$ .

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

Step 10.4.6

Combine the numerators over the common denominator.

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

Step 10.4.7

Add $1$ and $8$ .

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

Step 10.4.8

Cancel the common factor of $9$ and $3$ .

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Step 10.4.8.1

Factor $3$ out of $9$ .

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

Step 10.4.8.2

Cancel the common factors.

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Step 10.4.8.2.1

Factor $3$ out of $3$ .

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

Step 10.4.8.2.2

Cancel the common factor.

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

Step 10.4.8.2.3

Rewrite the expression.

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

Step 10.4.8.2.4

Divide $3$ by $1$ .

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

Step 10.4.9

Multiply $4$ by $3$ .

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

Step 10.4.10

One to any power is one.

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

Step 10.4.11

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

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

Step 10.4.12

Cancel the common factor of $4$ and $2$ .

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Step 10.4.12.1

Factor $2$ out of $4$ .

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

Step 10.4.12.2

Cancel the common factors.

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Step 10.4.12.2.1

Factor $2$ out of $2$ .

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

Step 10.4.12.2.2

Cancel the common factor.

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

Step 10.4.12.2.3

Rewrite the expression.

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

Step 10.4.12.2.4

Divide $2$ by $1$ .

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

Step 10.4.13

Multiply $- 1$ by $2$ .

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

Step 10.4.14

To write $- 2$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

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

Step 10.4.15

Combine $- 2$ and $\frac{2}{2}$ .

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

Step 10.4.16

Combine the numerators over the common denominator.

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

Step 10.4.17

Simplify the numerator.

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Step 10.4.17.1

Multiply $- 2$ by $2$ .

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

Step 10.4.17.2

Subtract $4$ from $1$ .

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

Step 10.4.18

Move the negative in front of the fraction.

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

Step 10.4.19

Multiply $- 1$ by $- 2$ .

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

Step 10.4.20

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

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

Step 10.4.21

Multiply $2$ by $3$ .

$12 + \frac{6}{2} + (3 \cdot 1) - 3 \cdot - 2$

Step 10.4.22

Cancel the common factor of $6$ and $2$ .

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Step 10.4.22.1

Factor $2$ out of $6$ .

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

Step 10.4.22.2

Cancel the common factors.

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Step 10.4.22.2.1

Factor $2$ out of $2$ .

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

Step 10.4.22.2.2

Cancel the common factor.

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

Step 10.4.22.2.3

Rewrite the expression.

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

Step 10.4.22.2.4

Divide $3$ by $1$ .

$12 + 3 + (3 \cdot 1) - 3 \cdot - 2$

Step 10.4.23

Add $12$ and $3$ .

$15 + (3 \cdot 1) - 3 \cdot - 2$

Step 10.4.24

Multiply $3$ by $1$ .

$15 + 3 - 3 \cdot - 2$

Step 10.4.25

Multiply $- 3$ by $- 2$ .

$15 + 3 + 6$

Step 10.4.26

Add $3$ and $6$ .

$15 + 9$

Step 10.4.27

Add $15$ and $9$ .

$24$

Step 11