Calculus Examples

$\int_{1}^{4} \frac{s^{4} - 8}{s^{2}} d s$

Step 1

Apply basic rules of exponents.

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Step 1.1

Move $s^{2}$ out of the denominator by raising it to the $- 1$ power.

$\int_{1}^{4} (s^{4} - 8) {(s^{2})}^{- 1} d s$

Step 1.2

Multiply the exponents in ${(s^{2})}^{- 1}$ .

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Step 1.2.1

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

$\int_{1}^{4} (s^{4} - 8) s^{2 \cdot - 1} d s$

Step 1.2.2

Multiply $2$ by $- 1$ .

$\int_{1}^{4} (s^{4} - 8) s^{- 2} d s$

Step 2

Multiply $(s^{4} - 8) s^{- 2}$ .

$\int_{1}^{4} s^{4} s^{- 2} - 8 s^{- 2} d s$

Step 3

Multiply $s^{4}$ by $s^{- 2}$ by adding the exponents.

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Step 3.1

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\int_{1}^{4} s^{4 - 2} - 8 s^{- 2} d s$

Step 3.2

Subtract $2$ from $4$ .

$\int_{1}^{4} s^{2} - 8 s^{- 2} d s$

Step 4

Split the single integral into multiple integrals.

$\int_{1}^{4} s^{2} d s + \int_{1}^{4} - 8 s^{- 2} d s$

Step 5

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

$\frac{1}{3} s^{3}]_{1}^{4} + \int_{1}^{4} - 8 s^{- 2} d s$

Step 6

Since $- 8$ is constant with respect to $s$ , move $- 8$ out of the integral.

$\frac{1}{3} s^{3}]_{1}^{4} - 8 \int_{1}^{4} s^{- 2} d s$

Step 7

By the Power Rule, the integral of $s^{- 2}$ with respect to $s$ is $- s^{- 1}$ .

$\frac{1}{3} s^{3}]_{1}^{4} - 8 (- s^{- 1}]_{1}^{4})$

Step 8

Substitute and simplify.

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Step 8.1

Evaluate $\frac{1}{3} s^{3}$ at $4$ and at $1$ .

$(\frac{1}{3} \cdot 4^{3}) - \frac{1}{3} \cdot 1^{3} - 8 (- s^{- 1}]_{1}^{4})$

Step 8.2

Evaluate $- s^{- 1}$ at $4$ and at $1$ .

$\frac{1}{3} \cdot 4^{3} - \frac{1}{3} \cdot 1^{3} - 8 (- 4^{- 1} + 1^{- 1})$

Step 8.3

Simplify.

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Step 8.3.1

Raise $4$ to the power of $3$ .

$\frac{1}{3} \cdot 64 - \frac{1}{3} \cdot 1^{3} - 8 (- 4^{- 1} + 1^{- 1})$

Step 8.3.2

Combine $\frac{1}{3}$ and $64$ .

$\frac{64}{3} - \frac{1}{3} \cdot 1^{3} - 8 (- 4^{- 1} + 1^{- 1})$

Step 8.3.3

One to any power is one.

$\frac{64}{3} - \frac{1}{3} \cdot 1 - 8 (- 4^{- 1} + 1^{- 1})$

Step 8.3.4

Multiply $- 1$ by $1$ .

$\frac{64}{3} - \frac{1}{3} - 8 (- 4^{- 1} + 1^{- 1})$

Step 8.3.5

Combine the numerators over the common denominator.

$\frac{64 - 1}{3} - 8 (- 4^{- 1} + 1^{- 1})$

Step 8.3.6

Subtract $1$ from $64$ .

$\frac{63}{3} - 8 (- 4^{- 1} + 1^{- 1})$

Step 8.3.7

Cancel the common factor of $63$ and $3$ .

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Step 8.3.7.1

Factor $3$ out of $63$ .

$\frac{3 \cdot 21}{3} - 8 (- 4^{- 1} + 1^{- 1})$

Step 8.3.7.2

Cancel the common factors.

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Step 8.3.7.2.1

Factor $3$ out of $3$ .

$\frac{3 \cdot 21}{3 (1)} - 8 (- 4^{- 1} + 1^{- 1})$

Step 8.3.7.2.2

Cancel the common factor.

$\frac{3 \cdot 21}{3 \cdot 1} - 8 (- 4^{- 1} + 1^{- 1})$

Step 8.3.7.2.3

Rewrite the expression.

$\frac{21}{1} - 8 (- 4^{- 1} + 1^{- 1})$

Step 8.3.7.2.4

Divide $21$ by $1$ .

$21 - 8 (- 4^{- 1} + 1^{- 1})$

Step 8.3.8

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

$21 - 8 (- \frac{1}{4} + 1^{- 1})$

Step 8.3.9

One to any power is one.

$21 - 8 (- \frac{1}{4} + 1)$

Step 8.3.10

Write $1$ as a fraction with a common denominator.

$21 - 8 (- \frac{1}{4} + \frac{4}{4})$

Step 8.3.11

Combine the numerators over the common denominator.

$21 - 8 \frac{- 1 + 4}{4}$

Step 8.3.12

Add $- 1$ and $4$ .

$21 - 8 (\frac{3}{4})$

Step 8.3.13

Combine $- 8$ and $\frac{3}{4}$ .

$21 + \frac{- 8 \cdot 3}{4}$

Step 8.3.14

Multiply $- 8$ by $3$ .

$21 + \frac{- 24}{4}$

Step 8.3.15

Cancel the common factor of $- 24$ and $4$ .

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Step 8.3.15.1

Factor $4$ out of $- 24$ .

$21 + \frac{4 \cdot - 6}{4}$

Step 8.3.15.2

Cancel the common factors.

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Step 8.3.15.2.1

Factor $4$ out of $4$ .

$21 + \frac{4 \cdot - 6}{4 (1)}$

Step 8.3.15.2.2

Cancel the common factor.

$21 + \frac{4 \cdot - 6}{4 \cdot 1}$

Step 8.3.15.2.3

Rewrite the expression.

$21 + \frac{- 6}{1}$

Step 8.3.15.2.4

Divide $- 6$ by $1$ .

$21 - 6$

Step 8.3.16

Subtract $6$ from $21$ .

$15$

Step 9