Calculus Examples

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

Step 1

Since $5$ is constant with respect to $x$ , move $5$ out of the integral.

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

Step 2

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

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

Step 3

Simplify the answer.

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Step 3.1

Simplify.

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Step 3.1.1

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

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

Step 3.1.2

Move $x^{- 3}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

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

Step 3.2

Substitute and simplify.

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Step 3.2.1

Evaluate $- \frac{1}{3 x^{3}}$ at $- 1$ and at $- 2$ .

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

Step 3.2.2

Simplify.

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Step 3.2.2.1

Raise $- 1$ to the power of $3$ .

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

Step 3.2.2.2

Multiply $3$ by $- 1$ .

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

Step 3.2.2.3

Move the negative in front of the fraction.

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

Step 3.2.2.4

Multiply $- 1$ by $- 1$ .

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

Step 3.2.2.5

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

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

Step 3.2.2.6

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

$5 (\frac{1}{3} + \frac{1}{3 \cdot - 8})$

Step 3.2.2.7

Multiply $3$ by $- 8$ .

$5 (\frac{1}{3} + \frac{1}{- 24})$

Step 3.2.2.8

Move the negative in front of the fraction.

$5 (\frac{1}{3} - \frac{1}{24})$

Step 3.2.2.9

To write $\frac{1}{3}$ as a fraction with a common denominator, multiply by $\frac{8}{8}$ .

$5 (\frac{1}{3} \cdot \frac{8}{8} - \frac{1}{24})$

Step 3.2.2.10

Write each expression with a common denominator of $24$ , by multiplying each by an appropriate factor of $1$ .

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Step 3.2.2.10.1

Multiply $\frac{1}{3}$ by $\frac{8}{8}$ .

$5 (\frac{8}{3 \cdot 8} - \frac{1}{24})$

Step 3.2.2.10.2

Multiply $3$ by $8$ .

$5 (\frac{8}{24} - \frac{1}{24})$

Step 3.2.2.11

Combine the numerators over the common denominator.

$5 \frac{8 - 1}{24}$

Step 3.2.2.12

Subtract $1$ from $8$ .

$5 (\frac{7}{24})$

Step 3.2.2.13

Combine $5$ and $\frac{7}{24}$ .

$\frac{5 \cdot 7}{24}$

Step 3.2.2.14

Multiply $5$ by $7$ .

$\frac{35}{24}$

Step 4

The result can be shown in multiple forms.

Exact Form:

$\frac{35}{24}$

Decimal Form:

$1.458\overline{3}$

Mixed Number Form:

$1 \frac{11}{24}$

Step 5