Calculus Examples

$\int_{1}^{4} \frac{t^{8} - t^{4}}{t^{6}} d t$

Step 1

Simplify the expression.

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Step 1.1

Simplify.

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Step 1.1.1

Factor $t^{4}$ out of $t^{8} - t^{4}$ .

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Step 1.1.1.1

Factor $t^{4}$ out of $t^{8}$ .

$\int_{1}^{4} \frac{t^{4} t^{4} - t^{4}}{t^{6}} d t$

Step 1.1.1.2

Factor $t^{4}$ out of $- t^{4}$ .

$\int_{1}^{4} \frac{t^{4} t^{4} + t^{4} \cdot - 1}{t^{6}} d t$

Step 1.1.1.3

Factor $t^{4}$ out of $t^{4} t^{4} + t^{4} \cdot - 1$ .

$\int_{1}^{4} \frac{t^{4} (t^{4} - 1)}{t^{6}} d t$

Step 1.1.2

Cancel the common factors.

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Step 1.1.2.1

Factor $t^{4}$ out of $t^{6}$ .

$\int_{1}^{4} \frac{t^{4} (t^{4} - 1)}{t^{4} t^{2}} d t$

Step 1.1.2.2

Cancel the common factor.

$\int_{1}^{4} \frac{t^{4} (t^{4} - 1)}{t^{4} t^{2}} d t$

Step 1.1.2.3

Rewrite the expression.

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

Step 1.2

Apply basic rules of exponents.

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Step 1.2.1

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

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

Step 1.2.2

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

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Step 1.2.2.1

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

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

Step 1.2.2.2

Multiply $2$ by $- 1$ .

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

Step 2

Multiply $(t^{4} - 1) t^{- 2}$ .

$\int_{1}^{4} t^{4} t^{- 2} - 1 t^{- 2} d t$

Step 3

Simplify.

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Step 3.1

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

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Step 3.1.1

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

$\int_{1}^{4} t^{4 - 2} - 1 t^{- 2} d t$

Step 3.1.2

Subtract $2$ from $4$ .

$\int_{1}^{4} t^{2} - 1 t^{- 2} d t$

Step 3.2

Rewrite $- 1 t^{- 2}$ as $- t^{- 2}$ .

$\int_{1}^{4} t^{2} - t^{- 2} d t$

Step 4

Split the single integral into multiple integrals.

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

Step 5

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

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

Step 6

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

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

Step 7

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

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

Step 8

Substitute and simplify.

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Step 8.1

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

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

Step 8.2

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

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

Step 8.3

Simplify.

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Step 8.3.1

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

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

Step 8.3.2

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

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

Step 8.3.3

One to any power is one.

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

Step 8.3.4

Multiply $- 1$ by $1$ .

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

Step 8.3.5

Combine the numerators over the common denominator.

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

Step 8.3.6

Subtract $1$ from $64$ .

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

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 $63$ .

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

Step 8.3.7.2

Cancel the common factors.

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Step 8.3.7.2.1

Factor $3$ out of $3$ .

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

Step 8.3.7.2.2

Cancel the common factor.

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

Step 8.3.7.2.3

Rewrite the expression.

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

Step 8.3.7.2.4

Divide $21$ by $1$ .

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

Step 8.3.8

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

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

Step 8.3.9

One to any power is one.

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

Step 8.3.10

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

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

Step 8.3.11

Combine the numerators over the common denominator.

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

Step 8.3.12

Add $- 1$ and $4$ .

$21 - \frac{3}{4}$

Step 8.3.13

To write $21$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

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

Step 8.3.14

Combine $21$ and $\frac{4}{4}$ .

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

Step 8.3.15

Combine the numerators over the common denominator.

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

Step 8.3.16

Simplify the numerator.

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Step 8.3.16.1

Multiply $21$ by $4$ .

$\frac{84 - 3}{4}$

Step 8.3.16.2

Subtract $3$ from $84$ .

$\frac{81}{4}$

Step 9

The result can be shown in multiple forms.

Exact Form:

$\frac{81}{4}$

Decimal Form:

$20.25$

Mixed Number Form:

$20 \frac{1}{4}$

Step 10