Calculus Examples

$\int_{2}^{5} 0.5 x^{2} + 3 x + 7 d x$

Step 1

Split the single integral into multiple integrals.

$\int_{2}^{5} 0.5 x^{2} d x + \int_{2}^{5} 3 x d x + \int_{2}^{5} 7 d x$

Step 2

Since $0.5$ is constant with respect to $x$ , move $0.5$ out of the integral.

$0.5 \int_{2}^{5} x^{2} d x + \int_{2}^{5} 3 x d x + \int_{2}^{5} 7 d x$

Step 3

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

$0.5 (\frac{1}{3} x^{3}]_{2}^{5}) + \int_{2}^{5} 3 x d x + \int_{2}^{5} 7 d x$

Step 4

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

$0.5 (\frac{x^{3}}{3}]_{2}^{5}) + \int_{2}^{5} 3 x d x + \int_{2}^{5} 7 d x$

Step 5

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

$0.5 (\frac{x^{3}}{3}]_{2}^{5}) + 3 \int_{2}^{5} x d x + \int_{2}^{5} 7 d x$

Step 6

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

$0.5 (\frac{x^{3}}{3}]_{2}^{5}) + 3 (\frac{1}{2} x^{2}]_{2}^{5}) + \int_{2}^{5} 7 d x$

Step 7

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

$0.5 (\frac{x^{3}}{3}]_{2}^{5}) + 3 (\frac{x^{2}}{2}]_{2}^{5}) + \int_{2}^{5} 7 d x$

Step 8

Apply the constant rule.

$0.5 (\frac{x^{3}}{3}]_{2}^{5}) + 3 (\frac{x^{2}}{2}]_{2}^{5}) + 7 x]_{2}^{5}$

Step 9

Substitute and simplify.

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Step 9.1

Evaluate $\frac{x^{3}}{3}$ at $5$ and at $2$ .

$0.5 ((\frac{5^{3}}{3}) - \frac{2^{3}}{3}) + 3 (\frac{x^{2}}{2}]_{2}^{5}) + 7 x]_{2}^{5}$

Step 9.2

Evaluate $\frac{x^{2}}{2}$ at $5$ and at $2$ .

$0.5 (\frac{5^{3}}{3} - \frac{2^{3}}{3}) + 3 (\frac{5^{2}}{2} - \frac{2^{2}}{2}) + 7 x]_{2}^{5}$

Step 9.3

Evaluate $7 x$ at $5$ and at $2$ .

$0.5 (\frac{5^{3}}{3} - \frac{2^{3}}{3}) + 3 (\frac{5^{2}}{2} - \frac{2^{2}}{2}) + (7 \cdot 5) - 7 \cdot 2$

Step 9.4

Simplify.

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Step 9.4.1

Raise $5$ to the power of $3$ .

$0.5 (\frac{125}{3} - \frac{2^{3}}{3}) + 3 (\frac{5^{2}}{2} - \frac{2^{2}}{2}) + (7 \cdot 5) - 7 \cdot 2$

Step 9.4.2

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

$0.5 (\frac{125}{3} - \frac{8}{3}) + 3 (\frac{5^{2}}{2} - \frac{2^{2}}{2}) + (7 \cdot 5) - 7 \cdot 2$

Step 9.4.3

Combine the numerators over the common denominator.

$0.5 \frac{125 - 8}{3} + 3 (\frac{5^{2}}{2} - \frac{2^{2}}{2}) + (7 \cdot 5) - 7 \cdot 2$

Step 9.4.4

Subtract $8$ from $125$ .

$0.5 (\frac{117}{3}) + 3 (\frac{5^{2}}{2} - \frac{2^{2}}{2}) + (7 \cdot 5) - 7 \cdot 2$

Step 9.4.5

Cancel the common factor of $117$ and $3$ .

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Step 9.4.5.1

Factor $3$ out of $117$ .

$0.5 \frac{3 \cdot 39}{3} + 3 (\frac{5^{2}}{2} - \frac{2^{2}}{2}) + (7 \cdot 5) - 7 \cdot 2$

Step 9.4.5.2

Cancel the common factors.

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Step 9.4.5.2.1

Factor $3$ out of $3$ .

$0.5 \frac{3 \cdot 39}{3 (1)} + 3 (\frac{5^{2}}{2} - \frac{2^{2}}{2}) + (7 \cdot 5) - 7 \cdot 2$

Step 9.4.5.2.2

Cancel the common factor.

$0.5 \frac{3 \cdot 39}{3 \cdot 1} + 3 (\frac{5^{2}}{2} - \frac{2^{2}}{2}) + (7 \cdot 5) - 7 \cdot 2$

Step 9.4.5.2.3

Rewrite the expression.

$0.5 (\frac{39}{1}) + 3 (\frac{5^{2}}{2} - \frac{2^{2}}{2}) + (7 \cdot 5) - 7 \cdot 2$

Step 9.4.5.2.4

Divide $39$ by $1$ .

$0.5 \cdot 39 + 3 (\frac{5^{2}}{2} - \frac{2^{2}}{2}) + (7 \cdot 5) - 7 \cdot 2$

Step 9.4.6

Multiply $0.5$ by $39$ .

$19.5 + 3 (\frac{5^{2}}{2} - \frac{2^{2}}{2}) + (7 \cdot 5) - 7 \cdot 2$

Step 9.4.7

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

$19.5 + 3 (\frac{25}{2} - \frac{2^{2}}{2}) + (7 \cdot 5) - 7 \cdot 2$

Step 9.4.8

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

$19.5 + 3 (\frac{25}{2} - \frac{4}{2}) + (7 \cdot 5) - 7 \cdot 2$

Step 9.4.9

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

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Step 9.4.9.1

Factor $2$ out of $4$ .

$19.5 + 3 (\frac{25}{2} - \frac{2 \cdot 2}{2}) + (7 \cdot 5) - 7 \cdot 2$

Step 9.4.9.2

Cancel the common factors.

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Step 9.4.9.2.1

Factor $2$ out of $2$ .

$19.5 + 3 (\frac{25}{2} - \frac{2 \cdot 2}{2 (1)}) + (7 \cdot 5) - 7 \cdot 2$

Step 9.4.9.2.2

Cancel the common factor.

$19.5 + 3 (\frac{25}{2} - \frac{2 \cdot 2}{2 \cdot 1}) + (7 \cdot 5) - 7 \cdot 2$

Step 9.4.9.2.3

Rewrite the expression.

$19.5 + 3 (\frac{25}{2} - \frac{2}{1}) + (7 \cdot 5) - 7 \cdot 2$

Step 9.4.9.2.4

Divide $2$ by $1$ .

$19.5 + 3 (\frac{25}{2} - 1 \cdot 2) + (7 \cdot 5) - 7 \cdot 2$

Step 9.4.10

Multiply $- 1$ by $2$ .

$19.5 + 3 (\frac{25}{2} - 2) + (7 \cdot 5) - 7 \cdot 2$

Step 9.4.11

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

$19.5 + 3 (\frac{25}{2} - 2 \cdot \frac{2}{2}) + (7 \cdot 5) - 7 \cdot 2$

Step 9.4.12

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

$19.5 + 3 (\frac{25}{2} + \frac{- 2 \cdot 2}{2}) + (7 \cdot 5) - 7 \cdot 2$

Step 9.4.13

Combine the numerators over the common denominator.

$19.5 + 3 \frac{25 - 2 \cdot 2}{2} + (7 \cdot 5) - 7 \cdot 2$

Step 9.4.14

Simplify the numerator.

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Step 9.4.14.1

Multiply $- 2$ by $2$ .

$19.5 + 3 \frac{25 - 4}{2} + (7 \cdot 5) - 7 \cdot 2$

Step 9.4.14.2

Subtract $4$ from $25$ .

$19.5 + 3 (\frac{21}{2}) + (7 \cdot 5) - 7 \cdot 2$

Step 9.4.15

Combine $3$ and $\frac{21}{2}$ .

$19.5 + \frac{3 \cdot 21}{2} + (7 \cdot 5) - 7 \cdot 2$

Step 9.4.16

Multiply $3$ by $21$ .

$19.5 + \frac{63}{2} + (7 \cdot 5) - 7 \cdot 2$

Step 9.4.17

To write $19.5$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$19.5 \cdot \frac{2}{2} + \frac{63}{2} + (7 \cdot 5) - 7 \cdot 2$

Step 9.4.18

Combine $19.5$ and $\frac{2}{2}$ .

$\frac{19.5 \cdot 2}{2} + \frac{63}{2} + (7 \cdot 5) - 7 \cdot 2$

Step 9.4.19

Combine the numerators over the common denominator.

$\frac{19.5 \cdot 2 + 63}{2} + (7 \cdot 5) - 7 \cdot 2$

Step 9.4.20

Simplify the numerator.

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Step 9.4.20.1

Multiply $19.5$ by $2$ .

$\frac{39 + 63}{2} + (7 \cdot 5) - 7 \cdot 2$

Step 9.4.20.2

Add $39$ and $63$ .

$\frac{102}{2} + (7 \cdot 5) - 7 \cdot 2$

Step 9.4.21

Cancel the common factor of $102$ and $2$ .

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Step 9.4.21.1

Factor $2$ out of $102$ .

$\frac{2 \cdot 51}{2} + (7 \cdot 5) - 7 \cdot 2$

Step 9.4.21.2

Cancel the common factors.

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Step 9.4.21.2.1

Factor $2$ out of $2$ .

$\frac{2 \cdot 51}{2 (1)} + (7 \cdot 5) - 7 \cdot 2$

Step 9.4.21.2.2

Cancel the common factor.

$\frac{2 \cdot 51}{2 \cdot 1} + (7 \cdot 5) - 7 \cdot 2$

Step 9.4.21.2.3

Rewrite the expression.

$\frac{51}{1} + (7 \cdot 5) - 7 \cdot 2$

Step 9.4.21.2.4

Divide $51$ by $1$ .

$51 + (7 \cdot 5) - 7 \cdot 2$

Step 9.4.22

Multiply $7$ by $5$ .

$51 + 35 - 7 \cdot 2$

Step 9.4.23

Multiply $- 7$ by $2$ .

$51 + 35 - 14$

Step 9.4.24

Subtract $14$ from $35$ .

$51 + 21$

Step 9.4.25

Add $51$ and $21$ .

$72$

Step 10