Calculus Examples

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

Step 1

Apply basic rules of exponents.

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Step 1.1

Move $x^{3}$ out of the denominator by raising it to the $- 1$ power.

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

Step 1.2

Multiply the exponents in ${(x^{3})}^{- 1}$ .

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Step 1.2.1

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

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

Step 1.2.2

Multiply $3$ by $- 1$ .

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

Step 2

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

$- \frac{1}{2} x^{- 2}]_{1}^{8}$

Step 3

Substitute and simplify.

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Step 3.1

Evaluate $- \frac{1}{2} x^{- 2}$ at $8$ and at $1$ .

$(- \frac{1}{2} \cdot 8^{- 2}) + \frac{1}{2} \cdot 1^{- 2}$

Step 3.2

Simplify.

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Step 3.2.1

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

$- \frac{1}{2} \cdot \frac{1}{8^{2}} + \frac{1}{2} \cdot 1^{- 2}$

Step 3.2.2

Raise $8$ to the power of $2$ .

$- \frac{1}{2} \cdot \frac{1}{64} + \frac{1}{2} \cdot 1^{- 2}$

Step 3.2.3

Multiply $\frac{1}{64}$ by $\frac{1}{2}$ .

$- \frac{1}{64 \cdot 2} + \frac{1}{2} \cdot 1^{- 2}$

Step 3.2.4

Multiply $64$ by $2$ .

$- \frac{1}{128} + \frac{1}{2} \cdot 1^{- 2}$

Step 3.2.5

One to any power is one.

$- \frac{1}{128} + \frac{1}{2} \cdot 1$

Step 3.2.6

Multiply $\frac{1}{2}$ by $1$ .

$- \frac{1}{128} + \frac{1}{2}$

Step 3.2.7

To write $\frac{1}{2}$ as a fraction with a common denominator, multiply by $\frac{64}{64}$ .

$- \frac{1}{128} + \frac{1}{2} \cdot \frac{64}{64}$

Step 3.2.8

Write each expression with a common denominator of $128$ , by multiplying each by an appropriate factor of $1$ .

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Step 3.2.8.1

Multiply $\frac{1}{2}$ by $\frac{64}{64}$ .

$- \frac{1}{128} + \frac{64}{2 \cdot 64}$

Step 3.2.8.2

Multiply $2$ by $64$ .

$- \frac{1}{128} + \frac{64}{128}$

Step 3.2.9

Combine the numerators over the common denominator.

$\frac{- 1 + 64}{128}$

Step 3.2.10

Add $- 1$ and $64$ .

$\frac{63}{128}$

Step 4

The result can be shown in multiple forms.

Exact Form:

$\frac{63}{128}$

Decimal Form:

$0.4921875$

Step 5