Calculus Examples

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

Step 1

Split the single integral into multiple integrals.

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

Step 2

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

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

Step 3

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

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

Step 4

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

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

Step 5

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

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

Step 6

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

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

Step 7

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

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

Step 8

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

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

Step 9

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

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

Step 10

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

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

Step 11

Apply the constant rule.

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

Step 12

Substitute and simplify.

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Step 12.1

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

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

Step 12.2

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

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

Step 12.3

Evaluate $\frac{x^{2}}{2}$ at $4$ and at $1$ .

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

Step 12.4

Evaluate $x$ at $4$ and at $1$ .

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

Step 12.5

Simplify.

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Step 12.5.1

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

$4 (\frac{256}{4} - \frac{1^{4}}{4}) - 2 (\frac{4^{3}}{3} - \frac{1^{3}}{3}) + 3 (\frac{4^{2}}{2} - \frac{1^{2}}{2}) + 4 - 1$

Step 12.5.2

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

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Step 12.5.2.1

Factor $4$ out of $256$ .

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

Step 12.5.2.2

Cancel the common factors.

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Step 12.5.2.2.1

Factor $4$ out of $4$ .

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

Step 12.5.2.2.2

Cancel the common factor.

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

Step 12.5.2.2.3

Rewrite the expression.

$4 (\frac{64}{1} - \frac{1^{4}}{4}) - 2 (\frac{4^{3}}{3} - \frac{1^{3}}{3}) + 3 (\frac{4^{2}}{2} - \frac{1^{2}}{2}) + 4 - 1$

Step 12.5.2.2.4

Divide $64$ by $1$ .

$4 (64 - \frac{1^{4}}{4}) - 2 (\frac{4^{3}}{3} - \frac{1^{3}}{3}) + 3 (\frac{4^{2}}{2} - \frac{1^{2}}{2}) + 4 - 1$

Step 12.5.3

One to any power is one.

$4 (64 - \frac{1}{4}) - 2 (\frac{4^{3}}{3} - \frac{1^{3}}{3}) + 3 (\frac{4^{2}}{2} - \frac{1^{2}}{2}) + 4 - 1$

Step 12.5.4

To write $64$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

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

Step 12.5.5

Combine $64$ and $\frac{4}{4}$ .

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

Step 12.5.6

Combine the numerators over the common denominator.

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

Step 12.5.7

Simplify the numerator.

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Step 12.5.7.1

Multiply $64$ by $4$ .

$4 \frac{256 - 1}{4} - 2 (\frac{4^{3}}{3} - \frac{1^{3}}{3}) + 3 (\frac{4^{2}}{2} - \frac{1^{2}}{2}) + 4 - 1$

Step 12.5.7.2

Subtract $1$ from $256$ .

$4 (\frac{255}{4}) - 2 (\frac{4^{3}}{3} - \frac{1^{3}}{3}) + 3 (\frac{4^{2}}{2} - \frac{1^{2}}{2}) + 4 - 1$

Step 12.5.8

Combine $4$ and $\frac{255}{4}$ .

$\frac{4 \cdot 255}{4} - 2 (\frac{4^{3}}{3} - \frac{1^{3}}{3}) + 3 (\frac{4^{2}}{2} - \frac{1^{2}}{2}) + 4 - 1$

Step 12.5.9

Multiply $4$ by $255$ .

$\frac{1020}{4} - 2 (\frac{4^{3}}{3} - \frac{1^{3}}{3}) + 3 (\frac{4^{2}}{2} - \frac{1^{2}}{2}) + 4 - 1$

Step 12.5.10

Cancel the common factor of $1020$ and $4$ .

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Step 12.5.10.1

Factor $4$ out of $1020$ .

$\frac{4 \cdot 255}{4} - 2 (\frac{4^{3}}{3} - \frac{1^{3}}{3}) + 3 (\frac{4^{2}}{2} - \frac{1^{2}}{2}) + 4 - 1$

Step 12.5.10.2

Cancel the common factors.

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Step 12.5.10.2.1

Factor $4$ out of $4$ .

$\frac{4 \cdot 255}{4 (1)} - 2 (\frac{4^{3}}{3} - \frac{1^{3}}{3}) + 3 (\frac{4^{2}}{2} - \frac{1^{2}}{2}) + 4 - 1$

Step 12.5.10.2.2

Cancel the common factor.

$\frac{4 \cdot 255}{4 \cdot 1} - 2 (\frac{4^{3}}{3} - \frac{1^{3}}{3}) + 3 (\frac{4^{2}}{2} - \frac{1^{2}}{2}) + 4 - 1$

Step 12.5.10.2.3

Rewrite the expression.

$\frac{255}{1} - 2 (\frac{4^{3}}{3} - \frac{1^{3}}{3}) + 3 (\frac{4^{2}}{2} - \frac{1^{2}}{2}) + 4 - 1$

Step 12.5.10.2.4

Divide $255$ by $1$ .

$255 - 2 (\frac{4^{3}}{3} - \frac{1^{3}}{3}) + 3 (\frac{4^{2}}{2} - \frac{1^{2}}{2}) + 4 - 1$

Step 12.5.11

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

$255 - 2 (\frac{64}{3} - \frac{1^{3}}{3}) + 3 (\frac{4^{2}}{2} - \frac{1^{2}}{2}) + 4 - 1$

Step 12.5.12

One to any power is one.

$255 - 2 (\frac{64}{3} - \frac{1}{3}) + 3 (\frac{4^{2}}{2} - \frac{1^{2}}{2}) + 4 - 1$

Step 12.5.13

Combine the numerators over the common denominator.

$255 - 2 \frac{64 - 1}{3} + 3 (\frac{4^{2}}{2} - \frac{1^{2}}{2}) + 4 - 1$

Step 12.5.14

Subtract $1$ from $64$ .

$255 - 2 (\frac{63}{3}) + 3 (\frac{4^{2}}{2} - \frac{1^{2}}{2}) + 4 - 1$

Step 12.5.15

Cancel the common factor of $63$ and $3$ .

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Step 12.5.15.1

Factor $3$ out of $63$ .

$255 - 2 \frac{3 \cdot 21}{3} + 3 (\frac{4^{2}}{2} - \frac{1^{2}}{2}) + 4 - 1$

Step 12.5.15.2

Cancel the common factors.

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Step 12.5.15.2.1

Factor $3$ out of $3$ .

$255 - 2 \frac{3 \cdot 21}{3 (1)} + 3 (\frac{4^{2}}{2} - \frac{1^{2}}{2}) + 4 - 1$

Step 12.5.15.2.2

Cancel the common factor.

$255 - 2 \frac{3 \cdot 21}{3 \cdot 1} + 3 (\frac{4^{2}}{2} - \frac{1^{2}}{2}) + 4 - 1$

Step 12.5.15.2.3

Rewrite the expression.

$255 - 2 (\frac{21}{1}) + 3 (\frac{4^{2}}{2} - \frac{1^{2}}{2}) + 4 - 1$

Step 12.5.15.2.4

Divide $21$ by $1$ .

$255 - 2 \cdot 21 + 3 (\frac{4^{2}}{2} - \frac{1^{2}}{2}) + 4 - 1$

Step 12.5.16

Multiply $- 2$ by $21$ .

$255 - 42 + 3 (\frac{4^{2}}{2} - \frac{1^{2}}{2}) + 4 - 1$

Step 12.5.17

Subtract $42$ from $255$ .

$213 + 3 (\frac{4^{2}}{2} - \frac{1^{2}}{2}) + 4 - 1$

Step 12.5.18

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

$213 + 3 (\frac{16}{2} - \frac{1^{2}}{2}) + 4 - 1$

Step 12.5.19

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

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Step 12.5.19.1

Factor $2$ out of $16$ .

$213 + 3 (\frac{2 \cdot 8}{2} - \frac{1^{2}}{2}) + 4 - 1$

Step 12.5.19.2

Cancel the common factors.

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Step 12.5.19.2.1

Factor $2$ out of $2$ .

$213 + 3 (\frac{2 \cdot 8}{2 (1)} - \frac{1^{2}}{2}) + 4 - 1$

Step 12.5.19.2.2

Cancel the common factor.

$213 + 3 (\frac{2 \cdot 8}{2 \cdot 1} - \frac{1^{2}}{2}) + 4 - 1$

Step 12.5.19.2.3

Rewrite the expression.

$213 + 3 (\frac{8}{1} - \frac{1^{2}}{2}) + 4 - 1$

Step 12.5.19.2.4

Divide $8$ by $1$ .

$213 + 3 (8 - \frac{1^{2}}{2}) + 4 - 1$

Step 12.5.20

One to any power is one.

$213 + 3 (8 - \frac{1}{2}) + 4 - 1$

Step 12.5.21

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

$213 + 3 (8 \cdot \frac{2}{2} - \frac{1}{2}) + 4 - 1$

Step 12.5.22

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

$213 + 3 (\frac{8 \cdot 2}{2} - \frac{1}{2}) + 4 - 1$

Step 12.5.23

Combine the numerators over the common denominator.

$213 + 3 \frac{8 \cdot 2 - 1}{2} + 4 - 1$

Step 12.5.24

Simplify the numerator.

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Step 12.5.24.1

Multiply $8$ by $2$ .

$213 + 3 \frac{16 - 1}{2} + 4 - 1$

Step 12.5.24.2

Subtract $1$ from $16$ .

$213 + 3 (\frac{15}{2}) + 4 - 1$

Step 12.5.25

Combine $3$ and $\frac{15}{2}$ .

$213 + \frac{3 \cdot 15}{2} + 4 - 1$

Step 12.5.26

Multiply $3$ by $15$ .

$213 + \frac{45}{2} + 4 - 1$

Step 12.5.27

To write $213$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$213 \cdot \frac{2}{2} + \frac{45}{2} + 4 - 1$

Step 12.5.28

Combine $213$ and $\frac{2}{2}$ .

$\frac{213 \cdot 2}{2} + \frac{45}{2} + 4 - 1$

Step 12.5.29

Combine the numerators over the common denominator.

$\frac{213 \cdot 2 + 45}{2} + 4 - 1$

Step 12.5.30

Simplify the numerator.

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Step 12.5.30.1

Multiply $213$ by $2$ .

$\frac{426 + 45}{2} + 4 - 1$

Step 12.5.30.2

Add $426$ and $45$ .

$\frac{471}{2} + 4 - 1$

Step 12.5.31

Subtract $1$ from $4$ .

$\frac{471}{2} + 3$

Step 12.5.32

To write $3$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{471}{2} + 3 \cdot \frac{2}{2}$

Step 12.5.33

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

$\frac{471}{2} + \frac{3 \cdot 2}{2}$

Step 12.5.34

Combine the numerators over the common denominator.

$\frac{471 + 3 \cdot 2}{2}$

Step 12.5.35

Simplify the numerator.

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Step 12.5.35.1

Multiply $3$ by $2$ .

$\frac{471 + 6}{2}$

Step 12.5.35.2

Add $471$ and $6$ .

$\frac{477}{2}$

Step 13

The result can be shown in multiple forms.

Exact Form:

$\frac{477}{2}$

Decimal Form:

$238.5$

Mixed Number Form:

$238 \frac{1}{2}$

Step 14