Calculus Examples

$\int_{0.5}^{4} x^{3} - 6 x^{2} + 9 x + 1 d x$

Step 1

Split the single integral into multiple integrals.

$\int_{0.5}^{4} x^{3} d x + \int_{0.5}^{4} - 6 x^{2} d x + \int_{0.5}^{4} 9 x d x + \int_{0.5}^{4} d x$

Step 2

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

$\frac{1}{4} x^{4}]_{0.5}^{4} + \int_{0.5}^{4} - 6 x^{2} d x + \int_{0.5}^{4} 9 x d x + \int_{0.5}^{4} d x$

Step 3

Since $- 6$ is constant with respect to $x$ , move $- 6$ out of the integral.

$\frac{1}{4} x^{4}]_{0.5}^{4} - 6 \int_{0.5}^{4} x^{2} d x + \int_{0.5}^{4} 9 x d x + \int_{0.5}^{4} d x$

Step 4

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

$\frac{1}{4} x^{4}]_{0.5}^{4} - 6 (\frac{1}{3} x^{3}]_{0.5}^{4}) + \int_{0.5}^{4} 9 x d x + \int_{0.5}^{4} d x$

Step 5

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

$\frac{1}{4} x^{4}]_{0.5}^{4} - 6 (\frac{x^{3}}{3}]_{0.5}^{4}) + \int_{0.5}^{4} 9 x d x + \int_{0.5}^{4} d x$

Step 6

Since $9$ is constant with respect to $x$ , move $9$ out of the integral.

$\frac{1}{4} x^{4}]_{0.5}^{4} - 6 (\frac{x^{3}}{3}]_{0.5}^{4}) + 9 \int_{0.5}^{4} x d x + \int_{0.5}^{4} d x$

Step 7

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

$\frac{1}{4} x^{4}]_{0.5}^{4} - 6 (\frac{x^{3}}{3}]_{0.5}^{4}) + 9 (\frac{1}{2} x^{2}]_{0.5}^{4}) + \int_{0.5}^{4} d x$

Step 8

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

$\frac{1}{4} x^{4}]_{0.5}^{4} - 6 (\frac{x^{3}}{3}]_{0.5}^{4}) + 9 (\frac{x^{2}}{2}]_{0.5}^{4}) + \int_{0.5}^{4} d x$

Step 9

Apply the constant rule.

$\frac{1}{4} x^{4}]_{0.5}^{4} - 6 (\frac{x^{3}}{3}]_{0.5}^{4}) + 9 (\frac{x^{2}}{2}]_{0.5}^{4}) + x]_{0.5}^{4}$

Step 10

Simplify the answer.

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Step 10.1

Combine $\frac{1}{4} x^{4}]_{0.5}^{4}$ and $x]_{0.5}^{4}$ .

$\frac{1}{4} x^{4} + x]_{0.5}^{4} - 6 (\frac{x^{3}}{3}]_{0.5}^{4}) + 9 (\frac{x^{2}}{2}]_{0.5}^{4})$

Step 10.2

Substitute and simplify.

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Step 10.2.1

Evaluate $\frac{1}{4} x^{4} + x$ at $4$ and at $0.5$ .

$(\frac{1}{4} \cdot 4^{4} + 4) - (\frac{1}{4} \cdot {0.5}^{4} + 0.5) - 6 (\frac{x^{3}}{3}]_{0.5}^{4}) + 9 (\frac{x^{2}}{2}]_{0.5}^{4})$

Step 10.2.2

Evaluate $\frac{x^{3}}{3}$ at $4$ and at $0.5$ .

$(\frac{1}{4} \cdot 4^{4} + 4) - (\frac{1}{4} \cdot {0.5}^{4} + 0.5) - 6 ((\frac{4^{3}}{3}) - \frac{{0.5}^{3}}{3}) + 9 (\frac{x^{2}}{2}]_{0.5}^{4})$

Step 10.2.3

Evaluate $\frac{x^{2}}{2}$ at $4$ and at $0.5$ .

$(\frac{1}{4} \cdot 4^{4} + 4) - (\frac{1}{4} \cdot {0.5}^{4} + 0.5) - 6 (\frac{4^{3}}{3} - \frac{{0.5}^{3}}{3}) + 9 (\frac{4^{2}}{2} - \frac{{0.5}^{2}}{2})$

Step 10.2.4

Simplify.

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Step 10.2.4.1

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

$\frac{1}{4} \cdot 256 + 4 - (\frac{1}{4} \cdot {0.5}^{4} + 0.5) - 6 (\frac{4^{3}}{3} - \frac{{0.5}^{3}}{3}) + 9 (\frac{4^{2}}{2} - \frac{{0.5}^{2}}{2})$

Step 10.2.4.2

Combine $\frac{1}{4}$ and $256$ .

$\frac{256}{4} + 4 - (\frac{1}{4} \cdot {0.5}^{4} + 0.5) - 6 (\frac{4^{3}}{3} - \frac{{0.5}^{3}}{3}) + 9 (\frac{4^{2}}{2} - \frac{{0.5}^{2}}{2})$

Step 10.2.4.3

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

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Step 10.2.4.3.1

Factor $4$ out of $256$ .

$\frac{4 \cdot 64}{4} + 4 - (\frac{1}{4} \cdot {0.5}^{4} + 0.5) - 6 (\frac{4^{3}}{3} - \frac{{0.5}^{3}}{3}) + 9 (\frac{4^{2}}{2} - \frac{{0.5}^{2}}{2})$

Step 10.2.4.3.2

Cancel the common factors.

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Step 10.2.4.3.2.1

Factor $4$ out of $4$ .

$\frac{4 \cdot 64}{4 (1)} + 4 - (\frac{1}{4} \cdot {0.5}^{4} + 0.5) - 6 (\frac{4^{3}}{3} - \frac{{0.5}^{3}}{3}) + 9 (\frac{4^{2}}{2} - \frac{{0.5}^{2}}{2})$

Step 10.2.4.3.2.2

Cancel the common factor.

$\frac{4 \cdot 64}{4 \cdot 1} + 4 - (\frac{1}{4} \cdot {0.5}^{4} + 0.5) - 6 (\frac{4^{3}}{3} - \frac{{0.5}^{3}}{3}) + 9 (\frac{4^{2}}{2} - \frac{{0.5}^{2}}{2})$

Step 10.2.4.3.2.3

Rewrite the expression.

$\frac{64}{1} + 4 - (\frac{1}{4} \cdot {0.5}^{4} + 0.5) - 6 (\frac{4^{3}}{3} - \frac{{0.5}^{3}}{3}) + 9 (\frac{4^{2}}{2} - \frac{{0.5}^{2}}{2})$

Step 10.2.4.3.2.4

Divide $64$ by $1$ .

$64 + 4 - (\frac{1}{4} \cdot {0.5}^{4} + 0.5) - 6 (\frac{4^{3}}{3} - \frac{{0.5}^{3}}{3}) + 9 (\frac{4^{2}}{2} - \frac{{0.5}^{2}}{2})$

Step 10.2.4.4

Add $64$ and $4$ .

$68 - (\frac{1}{4} \cdot {0.5}^{4} + 0.5) - 6 (\frac{4^{3}}{3} - \frac{{0.5}^{3}}{3}) + 9 (\frac{4^{2}}{2} - \frac{{0.5}^{2}}{2})$

Step 10.2.4.5

Raise $0.5$ to the power of $4$ .

$68 - (\frac{1}{4} \cdot 0.0625 + 0.5) - 6 (\frac{4^{3}}{3} - \frac{{0.5}^{3}}{3}) + 9 (\frac{4^{2}}{2} - \frac{{0.5}^{2}}{2})$

Step 10.2.4.6

Combine $\frac{1}{4}$ and $0.0625$ .

$68 - (\frac{0.0625}{4} + 0.5) - 6 (\frac{4^{3}}{3} - \frac{{0.5}^{3}}{3}) + 9 (\frac{4^{2}}{2} - \frac{{0.5}^{2}}{2})$

Step 10.2.4.7

To write $0.5$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$68 - (\frac{0.0625}{4} + 0.5 \cdot \frac{4}{4}) - 6 (\frac{4^{3}}{3} - \frac{{0.5}^{3}}{3}) + 9 (\frac{4^{2}}{2} - \frac{{0.5}^{2}}{2})$

Step 10.2.4.8

Combine $0.5$ and $\frac{4}{4}$ .

$68 - (\frac{0.0625}{4} + \frac{0.5 \cdot 4}{4}) - 6 (\frac{4^{3}}{3} - \frac{{0.5}^{3}}{3}) + 9 (\frac{4^{2}}{2} - \frac{{0.5}^{2}}{2})$

Step 10.2.4.9

Combine the numerators over the common denominator.

$68 - \frac{0.0625 + 0.5 \cdot 4}{4} - 6 (\frac{4^{3}}{3} - \frac{{0.5}^{3}}{3}) + 9 (\frac{4^{2}}{2} - \frac{{0.5}^{2}}{2})$

Step 10.2.4.10

Multiply $0.5$ by $4$ .

$68 - \frac{0.0625 + 2}{4} - 6 (\frac{4^{3}}{3} - \frac{{0.5}^{3}}{3}) + 9 (\frac{4^{2}}{2} - \frac{{0.5}^{2}}{2})$

Step 10.2.4.11

Add $0.0625$ and $2$ .

$68 - \frac{2.0625}{4} - 6 (\frac{4^{3}}{3} - \frac{{0.5}^{3}}{3}) + 9 (\frac{4^{2}}{2} - \frac{{0.5}^{2}}{2})$

Step 10.2.4.12

To write $68$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$68 \cdot \frac{4}{4} - \frac{2.0625}{4} - 6 (\frac{4^{3}}{3} - \frac{{0.5}^{3}}{3}) + 9 (\frac{4^{2}}{2} - \frac{{0.5}^{2}}{2})$

Step 10.2.4.13

Combine $68$ and $\frac{4}{4}$ .

$\frac{68 \cdot 4}{4} - \frac{2.0625}{4} - 6 (\frac{4^{3}}{3} - \frac{{0.5}^{3}}{3}) + 9 (\frac{4^{2}}{2} - \frac{{0.5}^{2}}{2})$

Step 10.2.4.14

Combine the numerators over the common denominator.

$\frac{68 \cdot 4 - 2.0625}{4} - 6 (\frac{4^{3}}{3} - \frac{{0.5}^{3}}{3}) + 9 (\frac{4^{2}}{2} - \frac{{0.5}^{2}}{2})$

Step 10.2.4.15

Multiply $68$ by $4$ .

$\frac{272 - 2.0625}{4} - 6 (\frac{4^{3}}{3} - \frac{{0.5}^{3}}{3}) + 9 (\frac{4^{2}}{2} - \frac{{0.5}^{2}}{2})$

Step 10.2.4.16

Subtract $2.0625$ from $272$ .

$\frac{269.9375}{4} - 6 (\frac{4^{3}}{3} - \frac{{0.5}^{3}}{3}) + 9 (\frac{4^{2}}{2} - \frac{{0.5}^{2}}{2})$

Step 10.2.4.17

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

$\frac{269.9375}{4} - 6 (\frac{64}{3} - \frac{{0.5}^{3}}{3}) + 9 (\frac{4^{2}}{2} - \frac{{0.5}^{2}}{2})$

Step 10.2.4.18

Raise $0.5$ to the power of $3$ .

$\frac{269.9375}{4} - 6 (\frac{64}{3} - \frac{0.125}{3}) + 9 (\frac{4^{2}}{2} - \frac{{0.5}^{2}}{2})$

Step 10.2.4.19

Combine the numerators over the common denominator.

$\frac{269.9375}{4} - 6 \frac{64 - 0.125}{3} + 9 (\frac{4^{2}}{2} - \frac{{0.5}^{2}}{2})$

Step 10.2.4.20

Subtract $0.125$ from $64$ .

$\frac{269.9375}{4} - 6 (\frac{63.875}{3}) + 9 (\frac{4^{2}}{2} - \frac{{0.5}^{2}}{2})$

Step 10.2.4.21

Combine $- 6$ and $\frac{63.875}{3}$ .

$\frac{269.9375}{4} + \frac{- 6 \cdot 63.875}{3} + 9 (\frac{4^{2}}{2} - \frac{{0.5}^{2}}{2})$

Step 10.2.4.22

Multiply $- 6$ by $63.875$ .

$\frac{269.9375}{4} + \frac{- 383.25}{3} + 9 (\frac{4^{2}}{2} - \frac{{0.5}^{2}}{2})$

Step 10.2.4.23

Move the negative in front of the fraction.

$\frac{269.9375}{4} - \frac{383.25}{3} + 9 (\frac{4^{2}}{2} - \frac{{0.5}^{2}}{2})$

Step 10.2.4.24

To write $\frac{269.9375}{4}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$\frac{269.9375}{4} \cdot \frac{3}{3} - \frac{383.25}{3} + 9 (\frac{4^{2}}{2} - \frac{{0.5}^{2}}{2})$

Step 10.2.4.25

To write $- \frac{383.25}{3}$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$\frac{269.9375}{4} \cdot \frac{3}{3} - \frac{383.25}{3} \cdot \frac{4}{4} + 9 (\frac{4^{2}}{2} - \frac{{0.5}^{2}}{2})$

Step 10.2.4.26

Write each expression with a common denominator of $12$ , by multiplying each by an appropriate factor of $1$ .

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Step 10.2.4.26.1

Multiply $\frac{269.9375}{4}$ by $\frac{3}{3}$ .

$\frac{269.9375 \cdot 3}{4 \cdot 3} - \frac{383.25}{3} \cdot \frac{4}{4} + 9 (\frac{4^{2}}{2} - \frac{{0.5}^{2}}{2})$

Step 10.2.4.26.2

Multiply $4$ by $3$ .

$\frac{269.9375 \cdot 3}{12} - \frac{383.25}{3} \cdot \frac{4}{4} + 9 (\frac{4^{2}}{2} - \frac{{0.5}^{2}}{2})$

Step 10.2.4.26.3

Multiply $\frac{383.25}{3}$ by $\frac{4}{4}$ .

$\frac{269.9375 \cdot 3}{12} - \frac{383.25 \cdot 4}{3 \cdot 4} + 9 (\frac{4^{2}}{2} - \frac{{0.5}^{2}}{2})$

Step 10.2.4.26.4

Multiply $3$ by $4$ .

$\frac{269.9375 \cdot 3}{12} - \frac{383.25 \cdot 4}{12} + 9 (\frac{4^{2}}{2} - \frac{{0.5}^{2}}{2})$

Step 10.2.4.27

Combine the numerators over the common denominator.

$\frac{269.9375 \cdot 3 - 383.25 \cdot 4}{12} + 9 (\frac{4^{2}}{2} - \frac{{0.5}^{2}}{2})$

Step 10.2.4.28

Multiply $269.9375$ by $3$ .

$\frac{809.8125 - 383.25 \cdot 4}{12} + 9 (\frac{4^{2}}{2} - \frac{{0.5}^{2}}{2})$

Step 10.2.4.29

Multiply $- 383.25$ by $4$ .

$\frac{809.8125 - 1533}{12} + 9 (\frac{4^{2}}{2} - \frac{{0.5}^{2}}{2})$

Step 10.2.4.30

Subtract $1533$ from $809.8125$ .

$\frac{- 723.1875}{12} + 9 (\frac{4^{2}}{2} - \frac{{0.5}^{2}}{2})$

Step 10.2.4.31

Move the negative in front of the fraction.

$- \frac{723.1875}{12} + 9 (\frac{4^{2}}{2} - \frac{{0.5}^{2}}{2})$

Step 10.2.4.32

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

$- \frac{723.1875}{12} + 9 (\frac{16}{2} - \frac{{0.5}^{2}}{2})$

Step 10.2.4.33

Cancel the common factor of $16$ and $2$ .

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Step 10.2.4.33.1

Factor $2$ out of $16$ .

$- \frac{723.1875}{12} + 9 (\frac{2 \cdot 8}{2} - \frac{{0.5}^{2}}{2})$

Step 10.2.4.33.2

Cancel the common factors.

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Step 10.2.4.33.2.1

Factor $2$ out of $2$ .

$- \frac{723.1875}{12} + 9 (\frac{2 \cdot 8}{2 (1)} - \frac{{0.5}^{2}}{2})$

Step 10.2.4.33.2.2

Cancel the common factor.

$- \frac{723.1875}{12} + 9 (\frac{2 \cdot 8}{2 \cdot 1} - \frac{{0.5}^{2}}{2})$

Step 10.2.4.33.2.3

Rewrite the expression.

$- \frac{723.1875}{12} + 9 (\frac{8}{1} - \frac{{0.5}^{2}}{2})$

Step 10.2.4.33.2.4

Divide $8$ by $1$ .

$- \frac{723.1875}{12} + 9 (8 - \frac{{0.5}^{2}}{2})$

Step 10.2.4.34

Raise $0.5$ to the power of $2$ .

$- \frac{723.1875}{12} + 9 (8 - \frac{0.25}{2})$

Step 10.2.4.35

To write $8$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$- \frac{723.1875}{12} + 9 (8 \cdot \frac{2}{2} - \frac{0.25}{2})$

Step 10.2.4.36

Combine $8$ and $\frac{2}{2}$ .

$- \frac{723.1875}{12} + 9 (\frac{8 \cdot 2}{2} - \frac{0.25}{2})$

Step 10.2.4.37

Combine the numerators over the common denominator.

$- \frac{723.1875}{12} + 9 \frac{8 \cdot 2 - 0.25}{2}$

Step 10.2.4.38

Multiply $8$ by $2$ .

$- \frac{723.1875}{12} + 9 \frac{16 - 0.25}{2}$

Step 10.2.4.39

Subtract $0.25$ from $16$ .

$- \frac{723.1875}{12} + 9 (\frac{15.75}{2})$

Step 10.2.4.40

Combine $9$ and $\frac{15.75}{2}$ .

$- \frac{723.1875}{12} + \frac{9 \cdot 15.75}{2}$

Step 10.2.4.41

Multiply $9$ by $15.75$ .

$- \frac{723.1875}{12} + \frac{141.75}{2}$

Step 10.2.4.42

To write $\frac{141.75}{2}$ as a fraction with a common denominator, multiply by $\frac{6}{6}$ .

$- \frac{723.1875}{12} + \frac{141.75}{2} \cdot \frac{6}{6}$

Step 10.2.4.43

Write each expression with a common denominator of $12$ , by multiplying each by an appropriate factor of $1$ .

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Step 10.2.4.43.1

Multiply $\frac{141.75}{2}$ by $\frac{6}{6}$ .

$- \frac{723.1875}{12} + \frac{141.75 \cdot 6}{2 \cdot 6}$

Step 10.2.4.43.2

Multiply $2$ by $6$ .

$- \frac{723.1875}{12} + \frac{141.75 \cdot 6}{12}$

Step 10.2.4.44

Combine the numerators over the common denominator.

$\frac{- 723.1875 + 141.75 \cdot 6}{12}$

Step 10.2.4.45

Multiply $141.75$ by $6$ .

$\frac{- 723.1875 + 850.5}{12}$

Step 10.2.4.46

Add $- 723.1875$ and $850.5$ .

$\frac{127.3125}{12}$

Step 11

Divide $127.3125$ by $12$ .

$10.609375$

Step 12