Calculus Examples

$\int_{- 2}^{1} (x^{4} + x^{3} - 4 x^{2} + 6) d x$

Step 1

Remove parentheses.

$\int_{- 2}^{1} x^{4} + x^{3} - 4 x^{2} + 6 d x$

Step 2

Split the single integral into multiple integrals.

$\int_{- 2}^{1} x^{4} d x + \int_{- 2}^{1} x^{3} d x + \int_{- 2}^{1} - 4 x^{2} d x + \int_{- 2}^{1} 6 d x$

Step 3

By the Power Rule, the integral of $x^{4}$ with respect to $x$ is $\frac{1}{5} x^{5}$ .

$\frac{1}{5} x^{5}]_{- 2}^{1} + \int_{- 2}^{1} x^{3} d x + \int_{- 2}^{1} - 4 x^{2} d x + \int_{- 2}^{1} 6 d x$

Step 4

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

$\frac{1}{5} x^{5}]_{- 2}^{1} + \frac{1}{4} x^{4}]_{- 2}^{1} + \int_{- 2}^{1} - 4 x^{2} d x + \int_{- 2}^{1} 6 d x$

Step 5

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

$\frac{1}{5} x^{5}]_{- 2}^{1} + \frac{1}{4} x^{4}]_{- 2}^{1} - 4 \int_{- 2}^{1} x^{2} d x + \int_{- 2}^{1} 6 d x$

Step 6

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

$\frac{1}{5} x^{5}]_{- 2}^{1} + \frac{1}{4} x^{4}]_{- 2}^{1} - 4 (\frac{1}{3} x^{3}]_{- 2}^{1}) + \int_{- 2}^{1} 6 d x$

Step 7

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

$\frac{1}{5} x^{5}]_{- 2}^{1} + \frac{1}{4} x^{4}]_{- 2}^{1} - 4 (\frac{x^{3}}{3}]_{- 2}^{1}) + \int_{- 2}^{1} 6 d x$

Step 8

Apply the constant rule.

$\frac{1}{5} x^{5}]_{- 2}^{1} + \frac{1}{4} x^{4}]_{- 2}^{1} - 4 (\frac{x^{3}}{3}]_{- 2}^{1}) + 6 x]_{- 2}^{1}$

Step 9

Simplify the answer.

Tap for more steps...

Step 9.1

Simplify.

Tap for more steps...

Step 9.1.1

Combine $\frac{1}{5} x^{5}]_{- 2}^{1}$ and $\frac{1}{4} x^{4}]_{- 2}^{1}$ .

$\frac{1}{5} x^{5} + \frac{1}{4} x^{4}]_{- 2}^{1} - 4 (\frac{x^{3}}{3}]_{- 2}^{1}) + 6 x]_{- 2}^{1}$

Step 9.1.2

Combine $\frac{1}{5} x^{5} + \frac{1}{4} x^{4}]_{- 2}^{1}$ and $6 x]_{- 2}^{1}$ .

$\frac{1}{5} x^{5} + \frac{1}{4} x^{4} + 6 x]_{- 2}^{1} - 4 (\frac{x^{3}}{3}]_{- 2}^{1})$

Step 9.2

Substitute and simplify.

Tap for more steps...

Step 9.2.1

Evaluate $\frac{1}{5} x^{5} + \frac{1}{4} x^{4} + 6 x$ at $1$ and at $- 2$ .

$(\frac{1}{5} \cdot 1^{5} + \frac{1}{4} \cdot 1^{4} + 6 \cdot 1) - (\frac{1}{5} {(- 2)}^{5} + \frac{1}{4} {(- 2)}^{4} + 6 \cdot - 2) - 4 (\frac{x^{3}}{3}]_{- 2}^{1})$

Step 9.2.2

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

$(\frac{1}{5} \cdot 1^{5} + \frac{1}{4} \cdot 1^{4} + 6 \cdot 1) - (\frac{1}{5} {(- 2)}^{5} + \frac{1}{4} {(- 2)}^{4} + 6 \cdot - 2) - 4 ((\frac{1^{3}}{3}) - \frac{{(- 2)}^{3}}{3})$

Step 9.2.3

Simplify.

Tap for more steps...

Step 9.2.3.1

One to any power is one.

$\frac{1}{5} \cdot 1 + \frac{1}{4} \cdot 1^{4} + 6 \cdot 1 - (\frac{1}{5} {(- 2)}^{5} + \frac{1}{4} {(- 2)}^{4} + 6 \cdot - 2) - 4 ((\frac{1^{3}}{3}) - \frac{{(- 2)}^{3}}{3})$

Step 9.2.3.2

Multiply $\frac{1}{5}$ by $1$ .

$\frac{1}{5} + \frac{1}{4} \cdot 1^{4} + 6 \cdot 1 - (\frac{1}{5} {(- 2)}^{5} + \frac{1}{4} {(- 2)}^{4} + 6 \cdot - 2) - 4 ((\frac{1^{3}}{3}) - \frac{{(- 2)}^{3}}{3})$

Step 9.2.3.3

One to any power is one.

$\frac{1}{5} + \frac{1}{4} \cdot 1 + 6 \cdot 1 - (\frac{1}{5} {(- 2)}^{5} + \frac{1}{4} {(- 2)}^{4} + 6 \cdot - 2) - 4 ((\frac{1^{3}}{3}) - \frac{{(- 2)}^{3}}{3})$

Step 9.2.3.4

Multiply $\frac{1}{4}$ by $1$ .

$\frac{1}{5} + \frac{1}{4} + 6 \cdot 1 - (\frac{1}{5} {(- 2)}^{5} + \frac{1}{4} {(- 2)}^{4} + 6 \cdot - 2) - 4 ((\frac{1^{3}}{3}) - \frac{{(- 2)}^{3}}{3})$

Step 9.2.3.5

To write $\frac{1}{5}$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$\frac{1}{5} \cdot \frac{4}{4} + \frac{1}{4} + 6 \cdot 1 - (\frac{1}{5} {(- 2)}^{5} + \frac{1}{4} {(- 2)}^{4} + 6 \cdot - 2) - 4 ((\frac{1^{3}}{3}) - \frac{{(- 2)}^{3}}{3})$

Step 9.2.3.6

To write $\frac{1}{4}$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$\frac{1}{5} \cdot \frac{4}{4} + \frac{1}{4} \cdot \frac{5}{5} + 6 \cdot 1 - (\frac{1}{5} {(- 2)}^{5} + \frac{1}{4} {(- 2)}^{4} + 6 \cdot - 2) - 4 ((\frac{1^{3}}{3}) - \frac{{(- 2)}^{3}}{3})$

Step 9.2.3.7

Write each expression with a common denominator of $20$ , by multiplying each by an appropriate factor of $1$ .

Tap for more steps...

Step 9.2.3.7.1

Multiply $\frac{1}{5}$ by $\frac{4}{4}$ .

$\frac{4}{5 \cdot 4} + \frac{1}{4} \cdot \frac{5}{5} + 6 \cdot 1 - (\frac{1}{5} {(- 2)}^{5} + \frac{1}{4} {(- 2)}^{4} + 6 \cdot - 2) - 4 ((\frac{1^{3}}{3}) - \frac{{(- 2)}^{3}}{3})$

Step 9.2.3.7.2

Multiply $5$ by $4$ .

$\frac{4}{20} + \frac{1}{4} \cdot \frac{5}{5} + 6 \cdot 1 - (\frac{1}{5} {(- 2)}^{5} + \frac{1}{4} {(- 2)}^{4} + 6 \cdot - 2) - 4 ((\frac{1^{3}}{3}) - \frac{{(- 2)}^{3}}{3})$

Step 9.2.3.7.3

Multiply $\frac{1}{4}$ by $\frac{5}{5}$ .

$\frac{4}{20} + \frac{5}{4 \cdot 5} + 6 \cdot 1 - (\frac{1}{5} {(- 2)}^{5} + \frac{1}{4} {(- 2)}^{4} + 6 \cdot - 2) - 4 ((\frac{1^{3}}{3}) - \frac{{(- 2)}^{3}}{3})$

Step 9.2.3.7.4

Multiply $4$ by $5$ .

$\frac{4}{20} + \frac{5}{20} + 6 \cdot 1 - (\frac{1}{5} {(- 2)}^{5} + \frac{1}{4} {(- 2)}^{4} + 6 \cdot - 2) - 4 ((\frac{1^{3}}{3}) - \frac{{(- 2)}^{3}}{3})$

Step 9.2.3.8

Combine the numerators over the common denominator.

$\frac{4 + 5}{20} + 6 \cdot 1 - (\frac{1}{5} {(- 2)}^{5} + \frac{1}{4} {(- 2)}^{4} + 6 \cdot - 2) - 4 ((\frac{1^{3}}{3}) - \frac{{(- 2)}^{3}}{3})$

Step 9.2.3.9

Add $4$ and $5$ .

$\frac{9}{20} + 6 \cdot 1 - (\frac{1}{5} {(- 2)}^{5} + \frac{1}{4} {(- 2)}^{4} + 6 \cdot - 2) - 4 ((\frac{1^{3}}{3}) - \frac{{(- 2)}^{3}}{3})$

Step 9.2.3.10

Multiply $6$ by $1$ .

$\frac{9}{20} + 6 - (\frac{1}{5} {(- 2)}^{5} + \frac{1}{4} {(- 2)}^{4} + 6 \cdot - 2) - 4 ((\frac{1^{3}}{3}) - \frac{{(- 2)}^{3}}{3})$

Step 9.2.3.11

To write $6$ as a fraction with a common denominator, multiply by $\frac{20}{20}$ .

$\frac{9}{20} + 6 \cdot \frac{20}{20} - (\frac{1}{5} {(- 2)}^{5} + \frac{1}{4} {(- 2)}^{4} + 6 \cdot - 2) - 4 ((\frac{1^{3}}{3}) - \frac{{(- 2)}^{3}}{3})$

Step 9.2.3.12

Combine $6$ and $\frac{20}{20}$ .

$\frac{9}{20} + \frac{6 \cdot 20}{20} - (\frac{1}{5} {(- 2)}^{5} + \frac{1}{4} {(- 2)}^{4} + 6 \cdot - 2) - 4 ((\frac{1^{3}}{3}) - \frac{{(- 2)}^{3}}{3})$

Step 9.2.3.13

Combine the numerators over the common denominator.

$\frac{9 + 6 \cdot 20}{20} - (\frac{1}{5} {(- 2)}^{5} + \frac{1}{4} {(- 2)}^{4} + 6 \cdot - 2) - 4 ((\frac{1^{3}}{3}) - \frac{{(- 2)}^{3}}{3})$

Step 9.2.3.14

Simplify the numerator.

Tap for more steps...

Step 9.2.3.14.1

Multiply $6$ by $20$ .

$\frac{9 + 120}{20} - (\frac{1}{5} {(- 2)}^{5} + \frac{1}{4} {(- 2)}^{4} + 6 \cdot - 2) - 4 ((\frac{1^{3}}{3}) - \frac{{(- 2)}^{3}}{3})$

Step 9.2.3.14.2

Add $9$ and $120$ .

$\frac{129}{20} - (\frac{1}{5} {(- 2)}^{5} + \frac{1}{4} {(- 2)}^{4} + 6 \cdot - 2) - 4 ((\frac{1^{3}}{3}) - \frac{{(- 2)}^{3}}{3})$

Step 9.2.3.15

Raise $- 2$ to the power of $5$ .

$\frac{129}{20} - (\frac{1}{5} \cdot - 32 + \frac{1}{4} {(- 2)}^{4} + 6 \cdot - 2) - 4 ((\frac{1^{3}}{3}) - \frac{{(- 2)}^{3}}{3})$

Step 9.2.3.16

Combine $\frac{1}{5}$ and $- 32$ .

$\frac{129}{20} - (\frac{- 32}{5} + \frac{1}{4} {(- 2)}^{4} + 6 \cdot - 2) - 4 ((\frac{1^{3}}{3}) - \frac{{(- 2)}^{3}}{3})$

Step 9.2.3.17

Move the negative in front of the fraction.

$\frac{129}{20} - (- \frac{32}{5} + \frac{1}{4} {(- 2)}^{4} + 6 \cdot - 2) - 4 ((\frac{1^{3}}{3}) - \frac{{(- 2)}^{3}}{3})$

Step 9.2.3.18

Raise $- 2$ to the power of $4$ .

$\frac{129}{20} - (- \frac{32}{5} + \frac{1}{4} \cdot 16 + 6 \cdot - 2) - 4 ((\frac{1^{3}}{3}) - \frac{{(- 2)}^{3}}{3})$

Step 9.2.3.19

Combine $\frac{1}{4}$ and $16$ .

$\frac{129}{20} - (- \frac{32}{5} + \frac{16}{4} + 6 \cdot - 2) - 4 ((\frac{1^{3}}{3}) - \frac{{(- 2)}^{3}}{3})$

Step 9.2.3.20

Cancel the common factor of $16$ and $4$ .

Tap for more steps...

Step 9.2.3.20.1

Factor $4$ out of $16$ .

$\frac{129}{20} - (- \frac{32}{5} + \frac{4 \cdot 4}{4} + 6 \cdot - 2) - 4 ((\frac{1^{3}}{3}) - \frac{{(- 2)}^{3}}{3})$

Step 9.2.3.20.2

Cancel the common factors.

Tap for more steps...

Step 9.2.3.20.2.1

Factor $4$ out of $4$ .

$\frac{129}{20} - (- \frac{32}{5} + \frac{4 \cdot 4}{4 (1)} + 6 \cdot - 2) - 4 ((\frac{1^{3}}{3}) - \frac{{(- 2)}^{3}}{3})$

Step 9.2.3.20.2.2

Cancel the common factor.

$\frac{129}{20} - (- \frac{32}{5} + \frac{4 \cdot 4}{4 \cdot 1} + 6 \cdot - 2) - 4 ((\frac{1^{3}}{3}) - \frac{{(- 2)}^{3}}{3})$

Step 9.2.3.20.2.3

Rewrite the expression.

$\frac{129}{20} - (- \frac{32}{5} + \frac{4}{1} + 6 \cdot - 2) - 4 ((\frac{1^{3}}{3}) - \frac{{(- 2)}^{3}}{3})$

Step 9.2.3.20.2.4

Divide $4$ by $1$ .

$\frac{129}{20} - (- \frac{32}{5} + 4 + 6 \cdot - 2) - 4 ((\frac{1^{3}}{3}) - \frac{{(- 2)}^{3}}{3})$

Step 9.2.3.21

To write $4$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$\frac{129}{20} - (- \frac{32}{5} + 4 \cdot \frac{5}{5} + 6 \cdot - 2) - 4 ((\frac{1^{3}}{3}) - \frac{{(- 2)}^{3}}{3})$

Step 9.2.3.22

Combine $4$ and $\frac{5}{5}$ .

$\frac{129}{20} - (- \frac{32}{5} + \frac{4 \cdot 5}{5} + 6 \cdot - 2) - 4 ((\frac{1^{3}}{3}) - \frac{{(- 2)}^{3}}{3})$

Step 9.2.3.23

Combine the numerators over the common denominator.

$\frac{129}{20} - (\frac{- 32 + 4 \cdot 5}{5} + 6 \cdot - 2) - 4 ((\frac{1^{3}}{3}) - \frac{{(- 2)}^{3}}{3})$

Step 9.2.3.24

Simplify the numerator.

Tap for more steps...

Step 9.2.3.24.1

Multiply $4$ by $5$ .

$\frac{129}{20} - (\frac{- 32 + 20}{5} + 6 \cdot - 2) - 4 ((\frac{1^{3}}{3}) - \frac{{(- 2)}^{3}}{3})$

Step 9.2.3.24.2

Add $- 32$ and $20$ .

$\frac{129}{20} - (\frac{- 12}{5} + 6 \cdot - 2) - 4 ((\frac{1^{3}}{3}) - \frac{{(- 2)}^{3}}{3})$

Step 9.2.3.25

Move the negative in front of the fraction.

$\frac{129}{20} - (- \frac{12}{5} + 6 \cdot - 2) - 4 ((\frac{1^{3}}{3}) - \frac{{(- 2)}^{3}}{3})$

Step 9.2.3.26

Multiply $6$ by $- 2$ .

$\frac{129}{20} - (- \frac{12}{5} - 12) - 4 ((\frac{1^{3}}{3}) - \frac{{(- 2)}^{3}}{3})$

Step 9.2.3.27

To write $- 12$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$\frac{129}{20} - (- \frac{12}{5} - 12 \cdot \frac{5}{5}) - 4 ((\frac{1^{3}}{3}) - \frac{{(- 2)}^{3}}{3})$

Step 9.2.3.28

Combine $- 12$ and $\frac{5}{5}$ .

$\frac{129}{20} - (- \frac{12}{5} + \frac{- 12 \cdot 5}{5}) - 4 ((\frac{1^{3}}{3}) - \frac{{(- 2)}^{3}}{3})$

Step 9.2.3.29

Combine the numerators over the common denominator.

$\frac{129}{20} - \frac{- 12 - 12 \cdot 5}{5} - 4 ((\frac{1^{3}}{3}) - \frac{{(- 2)}^{3}}{3})$

Step 9.2.3.30

Simplify the numerator.

Tap for more steps...

Step 9.2.3.30.1

Multiply $- 12$ by $5$ .

$\frac{129}{20} - \frac{- 12 - 60}{5} - 4 ((\frac{1^{3}}{3}) - \frac{{(- 2)}^{3}}{3})$

Step 9.2.3.30.2

Subtract $60$ from $- 12$ .

$\frac{129}{20} - \frac{- 72}{5} - 4 ((\frac{1^{3}}{3}) - \frac{{(- 2)}^{3}}{3})$

Step 9.2.3.31

Move the negative in front of the fraction.

$\frac{129}{20} - - \frac{72}{5} - 4 ((\frac{1^{3}}{3}) - \frac{{(- 2)}^{3}}{3})$

Step 9.2.3.32

Multiply $- 1$ by $- 1$ .

$\frac{129}{20} + 1 (\frac{72}{5}) - 4 ((\frac{1^{3}}{3}) - \frac{{(- 2)}^{3}}{3})$

Step 9.2.3.33

Multiply $\frac{72}{5}$ by $1$ .

$\frac{129}{20} + \frac{72}{5} - 4 ((\frac{1^{3}}{3}) - \frac{{(- 2)}^{3}}{3})$

Step 9.2.3.34

To write $\frac{72}{5}$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$\frac{129}{20} + \frac{72}{5} \cdot \frac{4}{4} - 4 ((\frac{1^{3}}{3}) - \frac{{(- 2)}^{3}}{3})$

Step 9.2.3.35

Write each expression with a common denominator of $20$ , by multiplying each by an appropriate factor of $1$ .

Tap for more steps...

Step 9.2.3.35.1

Multiply $\frac{72}{5}$ by $\frac{4}{4}$ .

$\frac{129}{20} + \frac{72 \cdot 4}{5 \cdot 4} - 4 ((\frac{1^{3}}{3}) - \frac{{(- 2)}^{3}}{3})$

Step 9.2.3.35.2

Multiply $5$ by $4$ .

$\frac{129}{20} + \frac{72 \cdot 4}{20} - 4 ((\frac{1^{3}}{3}) - \frac{{(- 2)}^{3}}{3})$

Step 9.2.3.36

Combine the numerators over the common denominator.

$\frac{129 + 72 \cdot 4}{20} - 4 ((\frac{1^{3}}{3}) - \frac{{(- 2)}^{3}}{3})$

Step 9.2.3.37

Simplify the numerator.

Tap for more steps...

Step 9.2.3.37.1

Multiply $72$ by $4$ .

$\frac{129 + 288}{20} - 4 ((\frac{1^{3}}{3}) - \frac{{(- 2)}^{3}}{3})$

Step 9.2.3.37.2

Add $129$ and $288$ .

$\frac{417}{20} - 4 ((\frac{1^{3}}{3}) - \frac{{(- 2)}^{3}}{3})$

Step 9.2.3.38

One to any power is one.

$\frac{417}{20} - 4 (\frac{1}{3} - \frac{{(- 2)}^{3}}{3})$

Step 9.2.3.39

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

$\frac{417}{20} - 4 (\frac{1}{3} - \frac{- 8}{3})$

Step 9.2.3.40

Move the negative in front of the fraction.

$\frac{417}{20} - 4 (\frac{1}{3} - - \frac{8}{3})$

Step 9.2.3.41

Multiply $- 1$ by $- 1$ .

$\frac{417}{20} - 4 (\frac{1}{3} + 1 (\frac{8}{3}))$

Step 9.2.3.42

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

$\frac{417}{20} - 4 (\frac{1}{3} + \frac{8}{3})$

Step 9.2.3.43

Combine the numerators over the common denominator.

$\frac{417}{20} - 4 \frac{1 + 8}{3}$

Step 9.2.3.44

Add $1$ and $8$ .

$\frac{417}{20} - 4 (\frac{9}{3})$

Step 9.2.3.45

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

Tap for more steps...

Step 9.2.3.45.1

Factor $3$ out of $9$ .

$\frac{417}{20} - 4 \frac{3 \cdot 3}{3}$

Step 9.2.3.45.2

Cancel the common factors.

Tap for more steps...

Step 9.2.3.45.2.1

Factor $3$ out of $3$ .

$\frac{417}{20} - 4 \frac{3 \cdot 3}{3 (1)}$

Step 9.2.3.45.2.2

Cancel the common factor.

$\frac{417}{20} - 4 \frac{3 \cdot 3}{3 \cdot 1}$

Step 9.2.3.45.2.3

Rewrite the expression.

$\frac{417}{20} - 4 (\frac{3}{1})$

Step 9.2.3.45.2.4

Divide $3$ by $1$ .

$\frac{417}{20} - 4 \cdot 3$

Step 9.2.3.46

Multiply $- 4$ by $3$ .

$\frac{417}{20} - 12$

Step 9.2.3.47

To write $- 12$ as a fraction with a common denominator, multiply by $\frac{20}{20}$ .

$\frac{417}{20} - 12 \cdot \frac{20}{20}$

Step 9.2.3.48

Combine $- 12$ and $\frac{20}{20}$ .

$\frac{417}{20} + \frac{- 12 \cdot 20}{20}$

Step 9.2.3.49

Combine the numerators over the common denominator.

$\frac{417 - 12 \cdot 20}{20}$

Step 9.2.3.50

Simplify the numerator.

Tap for more steps...

Step 9.2.3.50.1

Multiply $- 12$ by $20$ .

$\frac{417 - 240}{20}$

Step 9.2.3.50.2

Subtract $240$ from $417$ .

$\frac{177}{20}$

Step 10

The result can be shown in multiple forms.

Exact Form:

$\frac{177}{20}$

Decimal Form:

$8.85$

Mixed Number Form:

$8 \frac{17}{20}$

Step 11