Calculus Examples

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

Step 1

Apply basic rules of exponents.

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Step 1.1

Move $x^{4}$ out of the denominator by raising it to the $- 1$ power.

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

Step 1.2

Multiply the exponents in ${(x^{4})}^{- 1}$ .

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Step 1.2.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

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

Step 1.2.2

Multiply $4$ by $- 1$ .

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

Step 2

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

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

Step 3

Substitute and simplify.

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Step 3.1

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

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

Step 3.2

Simplify.

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Step 3.2.1

One to any power is one.

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

Step 3.2.2

Multiply $- 1$ by $1$ .

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

Step 3.2.3

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

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

Step 3.2.4

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

$- \frac{1}{3} + \frac{1}{3} \cdot \frac{1}{- 8}$

Step 3.2.5

Move the negative in front of the fraction.

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

Step 3.2.6

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

$- \frac{1}{3} - \frac{1}{3 \cdot 8}$

Step 3.2.7

Multiply $3$ by $8$ .

$- \frac{1}{3} - \frac{1}{24}$

Step 3.2.8

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

$- \frac{1}{3} \cdot \frac{8}{8} - \frac{1}{24}$

Step 3.2.9

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

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Step 3.2.9.1

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

$- \frac{8}{3 \cdot 8} - \frac{1}{24}$

Step 3.2.9.2

Multiply $3$ by $8$ .

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

Step 3.2.10

Combine the numerators over the common denominator.

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

Step 3.2.11

Subtract $1$ from $- 8$ .

$\frac{- 9}{24}$

Step 3.2.12

Cancel the common factor of $- 9$ and $24$ .

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Step 3.2.12.1

Factor $3$ out of $- 9$ .

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

Step 3.2.12.2

Cancel the common factors.

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Step 3.2.12.2.1

Factor $3$ out of $24$ .

$\frac{3 \cdot - 3}{3 \cdot 8}$

Step 3.2.12.2.2

Cancel the common factor.

$\frac{3 \cdot - 3}{3 \cdot 8}$

Step 3.2.12.2.3

Rewrite the expression.

$\frac{- 3}{8}$

Step 3.2.13

Move the negative in front of the fraction.

$- \frac{3}{8}$

Step 4

The result can be shown in multiple forms.

Exact Form:

$- \frac{3}{8}$

Decimal Form:

$- 0.375$

Step 5