Calculus Examples

$\int_{1}^{2} \frac{t^{8} - t^{4}}{t^{6}} d t$

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 $t^{8}$ .

$\int_{1}^{2} \frac{t^{4} t^{4} - t^{4}}{t^{6}} d t$

Step 1.1.1.2

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

$\int_{1}^{2} \frac{t^{4} t^{4} + t^{4} \cdot - 1}{t^{6}} d t$

Step 1.1.1.3

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

$\int_{1}^{2} \frac{t^{4} (t^{4} - 1)}{t^{6}} d t$

Step 1.1.2

Cancel the common factors.

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Step 1.1.2.1

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

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

Step 1.1.2.2

Cancel the common factor.

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

Step 1.1.2.3

Rewrite the expression.

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

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.

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

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, ${(a^{m})}^{n} = a^{m n}$ .

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

Step 1.2.2.2

Multiply $2$ by $- 1$ .

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

Step 2

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

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

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.

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

Step 3.1.2

Subtract $2$ from $4$ .

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

Step 3.2

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

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

Step 4

Split the single integral into multiple integrals.

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

Step 5

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

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

Step 6

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

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

Step 7

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

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

Step 8

Substitute and simplify.

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Step 8.1

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

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

Step 8.2

Evaluate $- t^{- 1}$ at $2$ and at $1$ .

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

Step 8.3

Simplify.

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Step 8.3.1

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

$\frac{1}{3} \cdot 8 - \frac{1}{3} \cdot 1^{3} - (- 2^{- 1} + 1^{- 1})$

Step 8.3.2

Combine $\frac{1}{3}$ and $8$ .

$\frac{8}{3} - \frac{1}{3} \cdot 1^{3} - (- 2^{- 1} + 1^{- 1})$

Step 8.3.3

One to any power is one.

$\frac{8}{3} - \frac{1}{3} \cdot 1 - (- 2^{- 1} + 1^{- 1})$

Step 8.3.4

Multiply $- 1$ by $1$ .

$\frac{8}{3} - \frac{1}{3} - (- 2^{- 1} + 1^{- 1})$

Step 8.3.5

Combine the numerators over the common denominator.

$\frac{8 - 1}{3} - (- 2^{- 1} + 1^{- 1})$

Step 8.3.6

Subtract $1$ from $8$ .

$\frac{7}{3} - (- 2^{- 1} + 1^{- 1})$

Step 8.3.7

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

$\frac{7}{3} - (- \frac{1}{2} + 1^{- 1})$

Step 8.3.8

One to any power is one.

$\frac{7}{3} - (- \frac{1}{2} + 1)$

Step 8.3.9

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

$\frac{7}{3} - (- \frac{1}{2} + \frac{2}{2})$

Step 8.3.10

Combine the numerators over the common denominator.

$\frac{7}{3} - \frac{- 1 + 2}{2}$

Step 8.3.11

Add $- 1$ and $2$ .

$\frac{7}{3} - \frac{1}{2}$

Step 8.3.12

To write $\frac{7}{3}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{7}{3} \cdot \frac{2}{2} - \frac{1}{2}$

Step 8.3.13

To write $- \frac{1}{2}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$\frac{7}{3} \cdot \frac{2}{2} - \frac{1}{2} \cdot \frac{3}{3}$

Step 8.3.14

Write each expression with a common denominator of $6$ , by multiplying each by an appropriate factor of $1$ .

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Step 8.3.14.1

Multiply $\frac{7}{3}$ by $\frac{2}{2}$ .

$\frac{7 \cdot 2}{3 \cdot 2} - \frac{1}{2} \cdot \frac{3}{3}$

Step 8.3.14.2

Multiply $3$ by $2$ .

$\frac{7 \cdot 2}{6} - \frac{1}{2} \cdot \frac{3}{3}$

Step 8.3.14.3

Multiply $\frac{1}{2}$ by $\frac{3}{3}$ .

$\frac{7 \cdot 2}{6} - \frac{3}{2 \cdot 3}$

Step 8.3.14.4

Multiply $2$ by $3$ .

$\frac{7 \cdot 2}{6} - \frac{3}{6}$

Step 8.3.15

Combine the numerators over the common denominator.

$\frac{7 \cdot 2 - 3}{6}$

Step 8.3.16

Simplify the numerator.

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Step 8.3.16.1

Multiply $7$ by $2$ .

$\frac{14 - 3}{6}$

Step 8.3.16.2

Subtract $3$ from $14$ .

$\frac{11}{6}$

Step 9

The result can be shown in multiple forms.

Exact Form:

$\frac{11}{6}$

Decimal Form:

$1.8\overline{3}$

Mixed Number Form:

$1 \frac{5}{6}$

Step 10