Calculus Examples

$\int_{- 0.5}^{0.5} - 0.0015 x^{2} + 0.0397 x + 0.26 d x$

Step 1

Split the single integral into multiple integrals.

$\int_{- 0.5}^{0.5} - 0.0015 x^{2} d x + \int_{- 0.5}^{0.5} 0.0397 x d x + \int_{- 0.5}^{0.5} 0.26 d x$

Step 2

Since $- 0.0015$ is constant with respect to $x$ , move $- 0.0015$ out of the integral.

$- 0.0015 \int_{- 0.5}^{0.5} x^{2} d x + \int_{- 0.5}^{0.5} 0.0397 x d x + \int_{- 0.5}^{0.5} 0.26 d x$

Step 3

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

$- 0.0015 (\frac{1}{3} x^{3}]_{- 0.5}^{0.5}) + \int_{- 0.5}^{0.5} 0.0397 x d x + \int_{- 0.5}^{0.5} 0.26 d x$

Step 4

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

$- 0.0015 (\frac{x^{3}}{3}]_{- 0.5}^{0.5}) + \int_{- 0.5}^{0.5} 0.0397 x d x + \int_{- 0.5}^{0.5} 0.26 d x$

Step 5

Since $0.0397$ is constant with respect to $x$ , move $0.0397$ out of the integral.

$- 0.0015 (\frac{x^{3}}{3}]_{- 0.5}^{0.5}) + 0.0397 \int_{- 0.5}^{0.5} x d x + \int_{- 0.5}^{0.5} 0.26 d x$

Step 6

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

$- 0.0015 (\frac{x^{3}}{3}]_{- 0.5}^{0.5}) + 0.0397 (\frac{1}{2} x^{2}]_{- 0.5}^{0.5}) + \int_{- 0.5}^{0.5} 0.26 d x$

Step 7

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

$- 0.0015 (\frac{x^{3}}{3}]_{- 0.5}^{0.5}) + 0.0397 (\frac{x^{2}}{2}]_{- 0.5}^{0.5}) + \int_{- 0.5}^{0.5} 0.26 d x$

Step 8

Apply the constant rule.

$- 0.0015 (\frac{x^{3}}{3}]_{- 0.5}^{0.5}) + 0.0397 (\frac{x^{2}}{2}]_{- 0.5}^{0.5}) + 0.26 x]_{- 0.5}^{0.5}$

Step 9

Substitute and simplify.

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Step 9.1

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

$- 0.0015 ((\frac{{0.5}^{3}}{3}) - \frac{{(- 0.5)}^{3}}{3}) + 0.0397 (\frac{x^{2}}{2}]_{- 0.5}^{0.5}) + 0.26 x]_{- 0.5}^{0.5}$

Step 9.2

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

$- 0.0015 (\frac{{0.5}^{3}}{3} - \frac{{(- 0.5)}^{3}}{3}) + 0.0397 (\frac{{0.5}^{2}}{2} - \frac{{(- 0.5)}^{2}}{2}) + 0.26 x]_{- 0.5}^{0.5}$

Step 9.3

Evaluate $0.26 x$ at $0.5$ and at $- 0.5$ .

$- 0.0015 (\frac{{0.5}^{3}}{3} - \frac{{(- 0.5)}^{3}}{3}) + 0.0397 (\frac{{0.5}^{2}}{2} - \frac{{(- 0.5)}^{2}}{2}) + (0.26 \cdot 0.5) - 0.26 \cdot - 0.5$

Step 9.4

Simplify.

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Step 9.4.1

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

$- 0.0015 (\frac{0.125}{3} - \frac{{(- 0.5)}^{3}}{3}) + 0.0397 (\frac{{0.5}^{2}}{2} - \frac{{(- 0.5)}^{2}}{2}) + (0.26 \cdot 0.5) - 0.26 \cdot - 0.5$

Step 9.4.2

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

$- 0.0015 (\frac{0.125}{3} - \frac{- 0.125}{3}) + 0.0397 (\frac{{0.5}^{2}}{2} - \frac{{(- 0.5)}^{2}}{2}) + (0.26 \cdot 0.5) - 0.26 \cdot - 0.5$

Step 9.4.3

Move the negative in front of the fraction.

$- 0.0015 (\frac{0.125}{3} - - \frac{0.125}{3}) + 0.0397 (\frac{{0.5}^{2}}{2} - \frac{{(- 0.5)}^{2}}{2}) + (0.26 \cdot 0.5) - 0.26 \cdot - 0.5$

Step 9.4.4

Multiply $- 1$ by $- 1$ .

$- 0.0015 (\frac{0.125}{3} + 1 (\frac{0.125}{3})) + 0.0397 (\frac{{0.5}^{2}}{2} - \frac{{(- 0.5)}^{2}}{2}) + (0.26 \cdot 0.5) - 0.26 \cdot - 0.5$

Step 9.4.5

Multiply $\frac{0.125}{3}$ by $1$ .

$- 0.0015 (\frac{0.125}{3} + \frac{0.125}{3}) + 0.0397 (\frac{{0.5}^{2}}{2} - \frac{{(- 0.5)}^{2}}{2}) + (0.26 \cdot 0.5) - 0.26 \cdot - 0.5$

Step 9.4.6

Combine the numerators over the common denominator.

$- 0.0015 \frac{0.125 + 0.125}{3} + 0.0397 (\frac{{0.5}^{2}}{2} - \frac{{(- 0.5)}^{2}}{2}) + (0.26 \cdot 0.5) - 0.26 \cdot - 0.5$

Step 9.4.7

Add $0.125$ and $0.125$ .

$- 0.0015 (\frac{0.25}{3}) + 0.0397 (\frac{{0.5}^{2}}{2} - \frac{{(- 0.5)}^{2}}{2}) + (0.26 \cdot 0.5) - 0.26 \cdot - 0.5$

Step 9.4.8

Combine $- 0.0015$ and $\frac{0.25}{3}$ .

$\frac{- 0.0015 \cdot 0.25}{3} + 0.0397 (\frac{{0.5}^{2}}{2} - \frac{{(- 0.5)}^{2}}{2}) + (0.26 \cdot 0.5) - 0.26 \cdot - 0.5$

Step 9.4.9

Multiply $- 0.0015$ by $0.25$ .

$\frac{- 0.000375}{3} + 0.0397 (\frac{{0.5}^{2}}{2} - \frac{{(- 0.5)}^{2}}{2}) + (0.26 \cdot 0.5) - 0.26 \cdot - 0.5$

Step 9.4.10

Move the negative in front of the fraction.

$- \frac{0.000375}{3} + 0.0397 (\frac{{0.5}^{2}}{2} - \frac{{(- 0.5)}^{2}}{2}) + (0.26 \cdot 0.5) - 0.26 \cdot - 0.5$

Step 9.4.11

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

$- \frac{0.000375}{3} + 0.0397 (\frac{0.25}{2} - \frac{{(- 0.5)}^{2}}{2}) + (0.26 \cdot 0.5) - 0.26 \cdot - 0.5$

Step 9.4.12

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

$- \frac{0.000375}{3} + 0.0397 (\frac{0.25}{2} - \frac{0.25}{2}) + (0.26 \cdot 0.5) - 0.26 \cdot - 0.5$

Step 9.4.13

Combine the numerators over the common denominator.

$- \frac{0.000375}{3} + 0.0397 \frac{0.25 - 0.25}{2} + (0.26 \cdot 0.5) - 0.26 \cdot - 0.5$

Step 9.4.14

Subtract $0.25$ from $0.25$ .

$- \frac{0.000375}{3} + 0.0397 (\frac{0}{2}) + (0.26 \cdot 0.5) - 0.26 \cdot - 0.5$

Step 9.4.15

Cancel the common factor of $0$ and $2$ .

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Step 9.4.15.1

Factor $2$ out of $0$ .

$- \frac{0.000375}{3} + 0.0397 \frac{2 \cdot 0}{2} + (0.26 \cdot 0.5) - 0.26 \cdot - 0.5$

Step 9.4.15.2

Cancel the common factors.

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Step 9.4.15.2.1

Factor $2$ out of $2$ .

$- \frac{0.000375}{3} + 0.0397 \frac{2 \cdot 0}{2 (1)} + (0.26 \cdot 0.5) - 0.26 \cdot - 0.5$

Step 9.4.15.2.2

Cancel the common factor.

$- \frac{0.000375}{3} + 0.0397 \frac{2 \cdot 0}{2 \cdot 1} + (0.26 \cdot 0.5) - 0.26 \cdot - 0.5$

Step 9.4.15.2.3

Rewrite the expression.

$- \frac{0.000375}{3} + 0.0397 (\frac{0}{1}) + (0.26 \cdot 0.5) - 0.26 \cdot - 0.5$

Step 9.4.15.2.4

Divide $0$ by $1$ .

$- \frac{0.000375}{3} + 0.0397 \cdot 0 + (0.26 \cdot 0.5) - 0.26 \cdot - 0.5$

Step 9.4.16

Multiply $0.0397$ by $0$ .

$- \frac{0.000375}{3} + 0 + (0.26 \cdot 0.5) - 0.26 \cdot - 0.5$

Step 9.4.17

Add $- \frac{0.000375}{3}$ and $0$ .

$- \frac{0.000375}{3} + (0.26 \cdot 0.5) - 0.26 \cdot - 0.5$

Step 9.4.18

Multiply $0.26$ by $0.5$ .

$- \frac{0.000375}{3} + 0.13 - 0.26 \cdot - 0.5$

Step 9.4.19

Multiply $- 0.26$ by $- 0.5$ .

$- \frac{0.000375}{3} + 0.13 + 0.13$

Step 9.4.20

Add $0.13$ and $0.13$ .

$- \frac{0.000375}{3} + 0.26$

Step 9.4.21

To write $0.26$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$- \frac{0.000375}{3} + 0.26 \cdot \frac{3}{3}$

Step 9.4.22

Combine $0.26$ and $\frac{3}{3}$ .

$- \frac{0.000375}{3} + \frac{0.26 \cdot 3}{3}$

Step 9.4.23

Combine the numerators over the common denominator.

$\frac{- 0.000375 + 0.26 \cdot 3}{3}$

Step 9.4.24

Multiply $0.26$ by $3$ .

$\frac{- 0.000375 + 0.78}{3}$

Step 9.4.25

Add $- 0.000375$ and $0.78$ .

$\frac{0.779625}{3}$

Step 10

Divide $0.779625$ by $3$ .

$0.259875$

Step 11