Calculus Examples

$\int_{3}^{9} - \frac{2}{t^{3}} d t$

Step 1

Since $- 1$ is constant with respect to $t$ , move $- 1$ out of the integral.

$- \int_{3}^{9} \frac{2}{t^{3}} d t$

Step 2

Since $2$ is constant with respect to $t$ , move $2$ out of the integral.

$- (2 \int_{3}^{9} \frac{1}{t^{3}} d t)$

Step 3

Simplify the expression.

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Step 3.1

Multiply $2$ by $- 1$ .

$- 2 \int_{3}^{9} \frac{1}{t^{3}} d t$

Step 3.2

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

$- 2 \int_{3}^{9} {(t^{3})}^{- 1} d t$

Step 3.3

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

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Step 3.3.1

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

$- 2 \int_{3}^{9} t^{3 \cdot - 1} d t$

Step 3.3.2

Multiply $3$ by $- 1$ .

$- 2 \int_{3}^{9} t^{- 3} d t$

Step 4

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

$- 2 (- \frac{1}{2} t^{- 2}]_{3}^{9})$

Step 5

Simplify the answer.

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Step 5.1

Simplify.

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Step 5.1.1

Combine $t^{- 2}$ and $\frac{1}{2}$ .

$- 2 (- \frac{t^{- 2}}{2}]_{3}^{9})$

Step 5.1.2

Move $t^{- 2}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$- 2 (- \frac{1}{2 t^{2}}]_{3}^{9})$

Step 5.2

Substitute and simplify.

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Step 5.2.1

Evaluate $- \frac{1}{2 t^{2}}$ at $9$ and at $3$ .

$- 2 ((- \frac{1}{2 \cdot 9^{2}}) + \frac{1}{2 \cdot 3^{2}})$

Step 5.2.2

Simplify.

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Step 5.2.2.1

Raise $9$ to the power of $2$ .

$- 2 (- \frac{1}{2 \cdot 81} + \frac{1}{2 \cdot 3^{2}})$

Step 5.2.2.2

Multiply $2$ by $81$ .

$- 2 (- \frac{1}{162} + \frac{1}{2 \cdot 3^{2}})$

Step 5.2.2.3

Raise $3$ to the power of $2$ .

$- 2 (- \frac{1}{162} + \frac{1}{2 \cdot 9})$

Step 5.2.2.4

Multiply $2$ by $9$ .

$- 2 (- \frac{1}{162} + \frac{1}{18})$

Step 5.2.2.5

To write $\frac{1}{18}$ as a fraction with a common denominator, multiply by $\frac{9}{9}$ .

$- 2 (- \frac{1}{162} + \frac{1}{18} \cdot \frac{9}{9})$

Step 5.2.2.6

Write each expression with a common denominator of $162$ , by multiplying each by an appropriate factor of $1$ .

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Step 5.2.2.6.1

Multiply $\frac{1}{18}$ by $\frac{9}{9}$ .

$- 2 (- \frac{1}{162} + \frac{9}{18 \cdot 9})$

Step 5.2.2.6.2

Multiply $18$ by $9$ .

$- 2 (- \frac{1}{162} + \frac{9}{162})$

Step 5.2.2.7

Combine the numerators over the common denominator.

$- 2 \frac{- 1 + 9}{162}$

Step 5.2.2.8

Add $- 1$ and $9$ .

$- 2 (\frac{8}{162})$

Step 5.2.2.9

Cancel the common factor of $8$ and $162$ .

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Step 5.2.2.9.1

Factor $2$ out of $8$ .

$- 2 \frac{2 (4)}{162}$

Step 5.2.2.9.2

Cancel the common factors.

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Step 5.2.2.9.2.1

Factor $2$ out of $162$ .

$- 2 \frac{2 \cdot 4}{2 \cdot 81}$

Step 5.2.2.9.2.2

Cancel the common factor.

$- 2 \frac{2 \cdot 4}{2 \cdot 81}$

Step 5.2.2.9.2.3

Rewrite the expression.

$- 2 (\frac{4}{81})$

Step 5.2.2.10

Combine $- 2$ and $\frac{4}{81}$ .

$\frac{- 2 \cdot 4}{81}$

Step 5.2.2.11

Multiply $- 2$ by $4$ .

$\frac{- 8}{81}$

Step 5.2.2.12

Move the negative in front of the fraction.

$- \frac{8}{81}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$- \frac{8}{81}$

Decimal Form:

$- 0.09876543 \dots$

Step 7