Calculus Examples

$\int_{1}^{8} \frac{8}{3 x^{5}} d x$

Step 1

Since $\frac{8}{3}$ is constant with respect to $x$ , move $\frac{8}{3}$ out of the integral.

$\frac{8}{3} \int_{1}^{8} \frac{1}{x^{5}} d x$

Step 2

Apply basic rules of exponents.

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Step 2.1

Move $x^{5}$ out of the denominator by raising it to the $- 1$ power.

$\frac{8}{3} \int_{1}^{8} {(x^{5})}^{- 1} d x$

Step 2.2

Multiply the exponents in ${(x^{5})}^{- 1}$ .

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Step 2.2.1

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

$\frac{8}{3} \int_{1}^{8} x^{5 \cdot - 1} d x$

Step 2.2.2

Multiply $5$ by $- 1$ .

$\frac{8}{3} \int_{1}^{8} x^{- 5} d x$

Step 3

By the Power Rule, the integral of $x^{- 5}$ with respect to $x$ is $- \frac{1}{4} x^{- 4}$ .

$\frac{8}{3} - \frac{1}{4} x^{- 4}]_{1}^{8}$

Step 4

Substitute and simplify.

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Step 4.1

Evaluate $- \frac{1}{4} x^{- 4}$ at $8$ and at $1$ .

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

Step 4.2

Simplify.

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Step 4.2.1

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

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

Step 4.2.2

Raise $8$ to the power of $4$ .

$\frac{8}{3} (- \frac{1}{4} \cdot \frac{1}{4096} + \frac{1}{4} \cdot 1^{- 4})$

Step 4.2.3

Multiply $\frac{1}{4096}$ by $\frac{1}{4}$ .

$\frac{8}{3} (- \frac{1}{4096 \cdot 4} + \frac{1}{4} \cdot 1^{- 4})$

Step 4.2.4

Multiply $4096$ by $4$ .

$\frac{8}{3} (- \frac{1}{16384} + \frac{1}{4} \cdot 1^{- 4})$

Step 4.2.5

One to any power is one.

$\frac{8}{3} (- \frac{1}{16384} + \frac{1}{4} \cdot 1)$

Step 4.2.6

Multiply $\frac{1}{4}$ by $1$ .

$\frac{8}{3} (- \frac{1}{16384} + \frac{1}{4})$

Step 4.2.7

To write $\frac{1}{4}$ as a fraction with a common denominator, multiply by $\frac{4096}{4096}$ .

$\frac{8}{3} (- \frac{1}{16384} + \frac{1}{4} \cdot \frac{4096}{4096})$

Step 4.2.8

Write each expression with a common denominator of $16384$ , by multiplying each by an appropriate factor of $1$ .

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Step 4.2.8.1

Multiply $\frac{1}{4}$ by $\frac{4096}{4096}$ .

$\frac{8}{3} (- \frac{1}{16384} + \frac{4096}{4 \cdot 4096})$

Step 4.2.8.2

Multiply $4$ by $4096$ .

$\frac{8}{3} (- \frac{1}{16384} + \frac{4096}{16384})$

Step 4.2.9

Combine the numerators over the common denominator.

$\frac{8}{3} \cdot \frac{- 1 + 4096}{16384}$

Step 4.2.10

Add $- 1$ and $4096$ .

$\frac{8}{3} \cdot \frac{4095}{16384}$

Step 4.2.11

Multiply $\frac{8}{3}$ by $\frac{4095}{16384}$ .

$\frac{8 \cdot 4095}{3 \cdot 16384}$

Step 4.2.12

Multiply $8$ by $4095$ .

$\frac{32760}{3 \cdot 16384}$

Step 4.2.13

Multiply $3$ by $16384$ .

$\frac{32760}{49152}$

Step 4.2.14

Cancel the common factor of $32760$ and $49152$ .

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Step 4.2.14.1

Factor $24$ out of $32760$ .

$\frac{24 (1365)}{49152}$

Step 4.2.14.2

Cancel the common factors.

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Step 4.2.14.2.1

Factor $24$ out of $49152$ .

$\frac{24 \cdot 1365}{24 \cdot 2048}$

Step 4.2.14.2.2

Cancel the common factor.

$\frac{24 \cdot 1365}{24 \cdot 2048}$

Step 4.2.14.2.3

Rewrite the expression.

$\frac{1365}{2048}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$\frac{1365}{2048}$

Decimal Form:

$0.66650390 \dots$

Step 6