Calculus Examples

$\int_{1}^{4} 5 - 2 x + 3 x^{2} d x$

Step 1

Split the single integral into multiple integrals.

$\int_{1}^{4} 5 d x + \int_{1}^{4} - 2 x d x + \int_{1}^{4} 3 x^{2} d x$

Step 2

Apply the constant rule.

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

Step 3

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

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

Step 4

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

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

Step 5

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

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

Step 6

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

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

Step 7

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

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

Step 8

Simplify the answer.

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Step 8.1

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

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

Step 8.2

Substitute and simplify.

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Step 8.2.1

Evaluate $5 x$ at $4$ and at $1$ .

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

Step 8.2.2

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

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

Step 8.2.3

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

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

Step 8.2.4

Simplify.

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Step 8.2.4.1

Multiply $5$ by $4$ .

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

Step 8.2.4.2

Multiply $- 5$ by $1$ .

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

Step 8.2.4.3

Subtract $5$ from $20$ .

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

Step 8.2.4.4

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

$15 - 2 (\frac{16}{2} - \frac{1^{2}}{2}) + 3 ((\frac{4^{3}}{3}) - \frac{1^{3}}{3})$

Step 8.2.4.5

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

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Step 8.2.4.5.1

Factor $2$ out of $16$ .

$15 - 2 (\frac{2 \cdot 8}{2} - \frac{1^{2}}{2}) + 3 ((\frac{4^{3}}{3}) - \frac{1^{3}}{3})$

Step 8.2.4.5.2

Cancel the common factors.

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Step 8.2.4.5.2.1

Factor $2$ out of $2$ .

$15 - 2 (\frac{2 \cdot 8}{2 (1)} - \frac{1^{2}}{2}) + 3 ((\frac{4^{3}}{3}) - \frac{1^{3}}{3})$

Step 8.2.4.5.2.2

Cancel the common factor.

$15 - 2 (\frac{2 \cdot 8}{2 \cdot 1} - \frac{1^{2}}{2}) + 3 ((\frac{4^{3}}{3}) - \frac{1^{3}}{3})$

Step 8.2.4.5.2.3

Rewrite the expression.

$15 - 2 (\frac{8}{1} - \frac{1^{2}}{2}) + 3 ((\frac{4^{3}}{3}) - \frac{1^{3}}{3})$

Step 8.2.4.5.2.4

Divide $8$ by $1$ .

$15 - 2 (8 - \frac{1^{2}}{2}) + 3 ((\frac{4^{3}}{3}) - \frac{1^{3}}{3})$

Step 8.2.4.6

One to any power is one.

$15 - 2 (8 - \frac{1}{2}) + 3 ((\frac{4^{3}}{3}) - \frac{1^{3}}{3})$

Step 8.2.4.7

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

$15 - 2 (8 \cdot \frac{2}{2} - \frac{1}{2}) + 3 ((\frac{4^{3}}{3}) - \frac{1^{3}}{3})$

Step 8.2.4.8

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

$15 - 2 (\frac{8 \cdot 2}{2} - \frac{1}{2}) + 3 ((\frac{4^{3}}{3}) - \frac{1^{3}}{3})$

Step 8.2.4.9

Combine the numerators over the common denominator.

$15 - 2 \frac{8 \cdot 2 - 1}{2} + 3 ((\frac{4^{3}}{3}) - \frac{1^{3}}{3})$

Step 8.2.4.10

Simplify the numerator.

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Step 8.2.4.10.1

Multiply $8$ by $2$ .

$15 - 2 \frac{16 - 1}{2} + 3 ((\frac{4^{3}}{3}) - \frac{1^{3}}{3})$

Step 8.2.4.10.2

Subtract $1$ from $16$ .

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

Step 8.2.4.11

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

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

Step 8.2.4.12

Multiply $- 2$ by $15$ .

$15 + \frac{- 30}{2} + 3 ((\frac{4^{3}}{3}) - \frac{1^{3}}{3})$

Step 8.2.4.13

Cancel the common factor of $- 30$ and $2$ .

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Step 8.2.4.13.1

Factor $2$ out of $- 30$ .

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

Step 8.2.4.13.2

Cancel the common factors.

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Step 8.2.4.13.2.1

Factor $2$ out of $2$ .

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

Step 8.2.4.13.2.2

Cancel the common factor.

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

Step 8.2.4.13.2.3

Rewrite the expression.

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

Step 8.2.4.13.2.4

Divide $- 15$ by $1$ .

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

Step 8.2.4.14

Subtract $15$ from $15$ .

$0 + 3 ((\frac{4^{3}}{3}) - \frac{1^{3}}{3})$

Step 8.2.4.15

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

$0 + 3 (\frac{64}{3} - \frac{1^{3}}{3})$

Step 8.2.4.16

One to any power is one.

$0 + 3 (\frac{64}{3} - \frac{1}{3})$

Step 8.2.4.17

Combine the numerators over the common denominator.

$0 + 3 \frac{64 - 1}{3}$

Step 8.2.4.18

Subtract $1$ from $64$ .

$0 + 3 (\frac{63}{3})$

Step 8.2.4.19

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

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Step 8.2.4.19.1

Factor $3$ out of $63$ .

$0 + 3 \frac{3 \cdot 21}{3}$

Step 8.2.4.19.2

Cancel the common factors.

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Step 8.2.4.19.2.1

Factor $3$ out of $3$ .

$0 + 3 \frac{3 \cdot 21}{3 (1)}$

Step 8.2.4.19.2.2

Cancel the common factor.

$0 + 3 \frac{3 \cdot 21}{3 \cdot 1}$

Step 8.2.4.19.2.3

Rewrite the expression.

$0 + 3 (\frac{21}{1})$

Step 8.2.4.19.2.4

Divide $21$ by $1$ .

$0 + 3 \cdot 21$

Step 8.2.4.20

Multiply $3$ by $21$ .

$0 + 63$

Step 8.2.4.21

Add $0$ and $63$ .

$63$

Step 9