Calculus Examples

$\int_{- 1}^{1} (\frac{27}{x^{4}} - 3) d x$

Step 1

Remove parentheses.

$\int_{- 1}^{1} \frac{27}{x^{4}} - 3 d x$

Step 2

Split the single integral into multiple integrals.

$\int_{- 1}^{1} \frac{27}{x^{4}} d x + \int_{- 1}^{1} - 3 d x$

Step 3

Since $27$ is constant with respect to $x$ , move $27$ out of the integral.

$27 \int_{- 1}^{1} \frac{1}{x^{4}} d x + \int_{- 1}^{1} - 3 d x$

Step 4

Apply basic rules of exponents.

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Step 4.1

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

$27 \int_{- 1}^{1} {(x^{4})}^{- 1} d x + \int_{- 1}^{1} - 3 d x$

Step 4.2

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

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Step 4.2.1

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

$27 \int_{- 1}^{1} x^{4 \cdot - 1} d x + \int_{- 1}^{1} - 3 d x$

Step 4.2.2

Multiply $4$ by $- 1$ .

$27 \int_{- 1}^{1} x^{- 4} d x + \int_{- 1}^{1} - 3 d x$

Step 5

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

$27 (- \frac{1}{3} x^{- 3}]_{- 1}^{1}) + \int_{- 1}^{1} - 3 d x$

Step 6

Simplify.

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Step 6.1

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

$27 (- \frac{x^{- 3}}{3}]_{- 1}^{1}) + \int_{- 1}^{1} - 3 d x$

Step 6.2

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

$27 (- \frac{1}{3 x^{3}}]_{- 1}^{1}) + \int_{- 1}^{1} - 3 d x$

Step 7

Apply the constant rule.

$27 (- \frac{1}{3 x^{3}}]_{- 1}^{1}) + - 3 x]_{- 1}^{1}$

Step 8

Substitute and simplify.

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Step 8.1

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

$27 ((- \frac{1}{3 \cdot 1^{3}}) + \frac{1}{3 {(- 1)}^{3}}) + - 3 x]_{- 1}^{1}$

Step 8.2

Evaluate $- 3 x$ at $1$ and at $- 1$ .

$27 ((- \frac{1}{3 \cdot 1^{3}}) + \frac{1}{3 {(- 1)}^{3}}) + (- 3 \cdot 1) + 3 \cdot - 1$

Step 8.3

Simplify.

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Step 8.3.1

One to any power is one.

$27 (- \frac{1}{3 \cdot 1} + \frac{1}{3 {(- 1)}^{3}}) + (- 3 \cdot 1) + 3 \cdot - 1$

Step 8.3.2

Multiply $3$ by $1$ .

$27 (- \frac{1}{3} + \frac{1}{3 {(- 1)}^{3}}) + (- 3 \cdot 1) + 3 \cdot - 1$

Step 8.3.3

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

$27 (- \frac{1}{3} + \frac{1}{3 \cdot - 1}) + (- 3 \cdot 1) + 3 \cdot - 1$

Step 8.3.4

Multiply $3$ by $- 1$ .

$27 (- \frac{1}{3} + \frac{1}{- 3}) + (- 3 \cdot 1) + 3 \cdot - 1$

Step 8.3.5

Move the negative in front of the fraction.

$27 (- \frac{1}{3} - \frac{1}{3}) + (- 3 \cdot 1) + 3 \cdot - 1$

Step 8.3.6

Subtract $\frac{1}{3}$ from $- \frac{1}{3}$ .

$27 (- 2 (\frac{1}{3})) + (- 3 \cdot 1) + 3 \cdot - 1$

Step 8.3.7

Combine $- 2$ and $\frac{1}{3}$ .

$27 (\frac{- 2}{3}) + (- 3 \cdot 1) + 3 \cdot - 1$

Step 8.3.8

Move the negative in front of the fraction.

$27 (- \frac{2}{3}) + (- 3 \cdot 1) + 3 \cdot - 1$

Step 8.3.9

Multiply $- 1$ by $27$ .

$- 27 (\frac{2}{3}) + (- 3 \cdot 1) + 3 \cdot - 1$

Step 8.3.10

Combine $- 27$ and $\frac{2}{3}$ .

$\frac{- 27 \cdot 2}{3} + (- 3 \cdot 1) + 3 \cdot - 1$

Step 8.3.11

Multiply $- 27$ by $2$ .

$\frac{- 54}{3} + (- 3 \cdot 1) + 3 \cdot - 1$

Step 8.3.12

Cancel the common factor of $- 54$ and $3$ .

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Step 8.3.12.1

Factor $3$ out of $- 54$ .

$\frac{3 \cdot - 18}{3} + (- 3 \cdot 1) + 3 \cdot - 1$

Step 8.3.12.2

Cancel the common factors.

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Step 8.3.12.2.1

Factor $3$ out of $3$ .

$\frac{3 \cdot - 18}{3 (1)} + (- 3 \cdot 1) + 3 \cdot - 1$

Step 8.3.12.2.2

Cancel the common factor.

$\frac{3 \cdot - 18}{3 \cdot 1} + (- 3 \cdot 1) + 3 \cdot - 1$

Step 8.3.12.2.3

Rewrite the expression.

$\frac{- 18}{1} + (- 3 \cdot 1) + 3 \cdot - 1$

Step 8.3.12.2.4

Divide $- 18$ by $1$ .

$- 18 + (- 3 \cdot 1) + 3 \cdot - 1$

Step 8.3.13

Multiply $- 3$ by $1$ .

$- 18 - 3 + 3 \cdot - 1$

Step 8.3.14

Multiply $3$ by $- 1$ .

$- 18 - 3 - 3$

Step 8.3.15

Subtract $3$ from $- 3$ .

$- 18 - 6$

Step 8.3.16

Subtract $6$ from $- 18$ .

$- 24$

Step 9