Calculus Examples

$\int_{4}^{8} \frac{1}{t^{4}} d t$

Step 1

Apply basic rules of exponents.

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Step 1.1

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

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

Step 1.2

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

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Step 1.2.1

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

$\int_{4}^{8} t^{4 \cdot - 1} d t$

Step 1.2.2

Multiply $4$ by $- 1$ .

$\int_{4}^{8} t^{- 4} d t$

Step 2

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

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

Step 3

Substitute and simplify.

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Step 3.1

Evaluate $- \frac{1}{3} t^{- 3}$ at $8$ and at $4$ .

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

Step 3.2

Simplify.

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Step 3.2.1

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

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

Step 3.2.2

Raise $8$ to the power of $3$ .

$- \frac{1}{3} \cdot \frac{1}{512} + \frac{1}{3} \cdot 4^{- 3}$

Step 3.2.3

Multiply $\frac{1}{512}$ by $\frac{1}{3}$ .

$- \frac{1}{512 \cdot 3} + \frac{1}{3} \cdot 4^{- 3}$

Step 3.2.4

Multiply $512$ by $3$ .

$- \frac{1}{1536} + \frac{1}{3} \cdot 4^{- 3}$

Step 3.2.5

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

$- \frac{1}{1536} + \frac{1}{3} \cdot \frac{1}{4^{3}}$

Step 3.2.6

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

$- \frac{1}{1536} + \frac{1}{3} \cdot \frac{1}{64}$

Step 3.2.7

Multiply $\frac{1}{3}$ by $\frac{1}{64}$ .

$- \frac{1}{1536} + \frac{1}{3 \cdot 64}$

Step 3.2.8

Multiply $3$ by $64$ .

$- \frac{1}{1536} + \frac{1}{192}$

Step 3.2.9

To write $\frac{1}{192}$ as a fraction with a common denominator, multiply by $\frac{8}{8}$ .

$- \frac{1}{1536} + \frac{1}{192} \cdot \frac{8}{8}$

Step 3.2.10

Write each expression with a common denominator of $1536$ , by multiplying each by an appropriate factor of $1$ .

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Step 3.2.10.1

Multiply $\frac{1}{192}$ by $\frac{8}{8}$ .

$- \frac{1}{1536} + \frac{8}{192 \cdot 8}$

Step 3.2.10.2

Multiply $192$ by $8$ .

$- \frac{1}{1536} + \frac{8}{1536}$

Step 3.2.11

Combine the numerators over the common denominator.

$\frac{- 1 + 8}{1536}$

Step 3.2.12

Add $- 1$ and $8$ .

$\frac{7}{1536}$

Step 4

The result can be shown in multiple forms.

Exact Form:

$\frac{7}{1536}$

Decimal Form:

$0.00455729 \dots$

Step 5