Calculus Examples

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

Step 1

Split the single integral into multiple integrals.

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

Step 2

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

$\frac{1}{4} t^{4}]_{- 1}^{1} + \int_{- 1}^{1} - 9 t d t$

Step 3

Since $- 9$ is constant with respect to $t$ , move $- 9$ out of the integral.

$\frac{1}{4} t^{4}]_{- 1}^{1} - 9 \int_{- 1}^{1} t d t$

Step 4

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

$\frac{1}{4} t^{4}]_{- 1}^{1} - 9 (\frac{1}{2} t^{2}]_{- 1}^{1})$

Step 5

Simplify the answer.

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Step 5.1

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

$\frac{1}{4} t^{4}]_{- 1}^{1} - 9 (\frac{t^{2}}{2}]_{- 1}^{1})$

Step 5.2

Substitute and simplify.

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Step 5.2.1

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

$(\frac{1}{4} \cdot 1^{4}) - \frac{1}{4} {(- 1)}^{4} - 9 (\frac{t^{2}}{2}]_{- 1}^{1})$

Step 5.2.2

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

$\frac{1}{4} \cdot 1^{4} - \frac{1}{4} {(- 1)}^{4} - 9 (\frac{1^{2}}{2} - \frac{{(- 1)}^{2}}{2})$

Step 5.2.3

Simplify.

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Step 5.2.3.1

One to any power is one.

$\frac{1}{4} \cdot 1 - \frac{1}{4} {(- 1)}^{4} - 9 (\frac{1^{2}}{2} - \frac{{(- 1)}^{2}}{2})$

Step 5.2.3.2

Multiply $\frac{1}{4}$ by $1$ .

$\frac{1}{4} - \frac{1}{4} {(- 1)}^{4} - 9 (\frac{1^{2}}{2} - \frac{{(- 1)}^{2}}{2})$

Step 5.2.3.3

Raise $- 1$ to the power of $4$ .

$\frac{1}{4} - \frac{1}{4} \cdot 1 - 9 (\frac{1^{2}}{2} - \frac{{(- 1)}^{2}}{2})$

Step 5.2.3.4

Multiply $- 1$ by $1$ .

$\frac{1}{4} - \frac{1}{4} - 9 (\frac{1^{2}}{2} - \frac{{(- 1)}^{2}}{2})$

Step 5.2.3.5

Combine the numerators over the common denominator.

$\frac{1 - 1}{4} - 9 (\frac{1^{2}}{2} - \frac{{(- 1)}^{2}}{2})$

Step 5.2.3.6

Subtract $1$ from $1$ .

$\frac{0}{4} - 9 (\frac{1^{2}}{2} - \frac{{(- 1)}^{2}}{2})$

Step 5.2.3.7

Cancel the common factor of $0$ and $4$ .

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Step 5.2.3.7.1

Factor $4$ out of $0$ .

$\frac{4 (0)}{4} - 9 (\frac{1^{2}}{2} - \frac{{(- 1)}^{2}}{2})$

Step 5.2.3.7.2

Cancel the common factors.

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Step 5.2.3.7.2.1

Factor $4$ out of $4$ .

$\frac{4 \cdot 0}{4 \cdot 1} - 9 (\frac{1^{2}}{2} - \frac{{(- 1)}^{2}}{2})$

Step 5.2.3.7.2.2

Cancel the common factor.

$\frac{4 \cdot 0}{4 \cdot 1} - 9 (\frac{1^{2}}{2} - \frac{{(- 1)}^{2}}{2})$

Step 5.2.3.7.2.3

Rewrite the expression.

$\frac{0}{1} - 9 (\frac{1^{2}}{2} - \frac{{(- 1)}^{2}}{2})$

Step 5.2.3.7.2.4

Divide $0$ by $1$ .

$0 - 9 (\frac{1^{2}}{2} - \frac{{(- 1)}^{2}}{2})$

Step 5.2.3.8

One to any power is one.

$0 - 9 (\frac{1}{2} - \frac{{(- 1)}^{2}}{2})$

Step 5.2.3.9

Raise $- 1$ to the power of $2$ .

$0 - 9 (\frac{1}{2} - \frac{1}{2})$

Step 5.2.3.10

Combine the numerators over the common denominator.

$0 - 9 \frac{1 - 1}{2}$

Step 5.2.3.11

Subtract $1$ from $1$ .

$0 - 9 (\frac{0}{2})$

Step 5.2.3.12

Cancel the common factor of $0$ and $2$ .

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Step 5.2.3.12.1

Factor $2$ out of $0$ .

$0 - 9 \frac{2 (0)}{2}$

Step 5.2.3.12.2

Cancel the common factors.

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Step 5.2.3.12.2.1

Factor $2$ out of $2$ .

$0 - 9 \frac{2 \cdot 0}{2 \cdot 1}$

Step 5.2.3.12.2.2

Cancel the common factor.

$0 - 9 \frac{2 \cdot 0}{2 \cdot 1}$

Step 5.2.3.12.2.3

Rewrite the expression.

$0 - 9 (\frac{0}{1})$

Step 5.2.3.12.2.4

Divide $0$ by $1$ .

$0 - 9 \cdot 0$

Step 5.2.3.13

Multiply $- 9$ by $0$ .

$0 + 0$

Step 5.2.3.14

Add $0$ and $0$ .

$0$

Step 6