Calculus Examples

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

Step 1

Remove parentheses.

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

Step 2

Split the single integral into multiple integrals.

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

Step 3

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

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

Step 4

Apply the constant rule.

$- \frac{1}{2} x^{- 2}]_{\frac{1}{2}}^{1} + - 5 x]_{\frac{1}{2}}^{1}$

Step 5

Simplify the answer.

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Step 5.1

Combine $- \frac{1}{2} x^{- 2}]_{\frac{1}{2}}^{1}$ and $- 5 x]_{\frac{1}{2}}^{1}$ .

$- \frac{1}{2} x^{- 2} - 5 x]_{\frac{1}{2}}^{1}$

Step 5.2

Substitute and simplify.

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Step 5.2.1

Evaluate $- \frac{1}{2} x^{- 2} - 5 x$ at $1$ and at $\frac{1}{2}$ .

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

Step 5.2.2

Simplify.

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Step 5.2.2.1

One to any power is one.

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

Step 5.2.2.2

Multiply $- 1$ by $1$ .

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

Step 5.2.2.3

Multiply $- 5$ by $1$ .

$- \frac{1}{2} - 5 - (- \frac{1}{2} {(\frac{1}{2})}^{- 2} - 5 (\frac{1}{2}))$

Step 5.2.2.4

To write $- 5$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

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

Step 5.2.2.5

Combine $- 5$ and $\frac{2}{2}$ .

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

Step 5.2.2.6

Combine the numerators over the common denominator.

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

Step 5.2.2.7

Simplify the numerator.

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Step 5.2.2.7.1

Multiply $- 5$ by $2$ .

$\frac{- 1 - 10}{2} - (- \frac{1}{2} {(\frac{1}{2})}^{- 2} - 5 (\frac{1}{2}))$

Step 5.2.2.7.2

Subtract $10$ from $- 1$ .

$\frac{- 11}{2} - (- \frac{1}{2} {(\frac{1}{2})}^{- 2} - 5 (\frac{1}{2}))$

Step 5.2.2.8

Move the negative in front of the fraction.

$- \frac{11}{2} - (- \frac{1}{2} {(\frac{1}{2})}^{- 2} - 5 (\frac{1}{2}))$

Step 5.2.2.9

Change the sign of the exponent by rewriting the base as its reciprocal.

$- \frac{11}{2} - (- \frac{1}{2} \cdot 2^{2} - 5 (\frac{1}{2}))$

Step 5.2.2.10

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

$- \frac{11}{2} - (- \frac{1}{2} \cdot 4 - 5 (\frac{1}{2}))$

Step 5.2.2.11

Multiply $4$ by $- 1$ .

$- \frac{11}{2} - (- 4 (\frac{1}{2}) - 5 (\frac{1}{2}))$

Step 5.2.2.12

Combine $- 4$ and $\frac{1}{2}$ .

$- \frac{11}{2} - (\frac{- 4}{2} - 5 (\frac{1}{2}))$

Step 5.2.2.13

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

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Step 5.2.2.13.1

Factor $2$ out of $- 4$ .

$- \frac{11}{2} - (\frac{2 \cdot - 2}{2} - 5 (\frac{1}{2}))$

Step 5.2.2.13.2

Cancel the common factors.

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Step 5.2.2.13.2.1

Factor $2$ out of $2$ .

$- \frac{11}{2} - (\frac{2 \cdot - 2}{2 (1)} - 5 (\frac{1}{2}))$

Step 5.2.2.13.2.2

Cancel the common factor.

$- \frac{11}{2} - (\frac{2 \cdot - 2}{2 \cdot 1} - 5 (\frac{1}{2}))$

Step 5.2.2.13.2.3

Rewrite the expression.

$- \frac{11}{2} - (\frac{- 2}{1} - 5 (\frac{1}{2}))$

Step 5.2.2.13.2.4

Divide $- 2$ by $1$ .

$- \frac{11}{2} - (- 2 - 5 (\frac{1}{2}))$

Step 5.2.2.14

Combine $- 5$ and $\frac{1}{2}$ .

$- \frac{11}{2} - (- 2 + \frac{- 5}{2})$

Step 5.2.2.15

Move the negative in front of the fraction.

$- \frac{11}{2} - (- 2 - \frac{5}{2})$

Step 5.2.2.16

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

$- \frac{11}{2} - (- 2 \cdot \frac{2}{2} - \frac{5}{2})$

Step 5.2.2.17

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

$- \frac{11}{2} - (\frac{- 2 \cdot 2}{2} - \frac{5}{2})$

Step 5.2.2.18

Combine the numerators over the common denominator.

$- \frac{11}{2} - \frac{- 2 \cdot 2 - 5}{2}$

Step 5.2.2.19

Simplify the numerator.

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Step 5.2.2.19.1

Multiply $- 2$ by $2$ .

$- \frac{11}{2} - \frac{- 4 - 5}{2}$

Step 5.2.2.19.2

Subtract $5$ from $- 4$ .

$- \frac{11}{2} - \frac{- 9}{2}$

Step 5.2.2.20

Move the negative in front of the fraction.

$- \frac{11}{2} - - \frac{9}{2}$

Step 5.2.2.21

Multiply $- 1$ by $- 1$ .

$- \frac{11}{2} + 1 (\frac{9}{2})$

Step 5.2.2.22

Multiply $\frac{9}{2}$ by $1$ .

$- \frac{11}{2} + \frac{9}{2}$

Step 5.2.2.23

Combine the numerators over the common denominator.

$\frac{- 11 + 9}{2}$

Step 5.2.2.24

Add $- 11$ and $9$ .

$\frac{- 2}{2}$

Step 5.2.2.25

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

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Step 5.2.2.25.1

Factor $2$ out of $- 2$ .

$\frac{2 \cdot - 1}{2}$

Step 5.2.2.25.2

Cancel the common factors.

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Step 5.2.2.25.2.1

Factor $2$ out of $2$ .

$\frac{2 \cdot - 1}{2 (1)}$

Step 5.2.2.25.2.2

Cancel the common factor.

$\frac{2 \cdot - 1}{2 \cdot 1}$

Step 5.2.2.25.2.3

Rewrite the expression.

$\frac{- 1}{1}$

Step 5.2.2.25.2.4

Divide $- 1$ by $1$ .

$- 1$

Step 6