Calculus Examples

$\int_{0}^{1} 92 - 2 x^{2} - \frac{10}{3} x d x$

Step 1

Split the single integral into multiple integrals.

$\int_{0}^{1} 92 d x + \int_{0}^{1} - 2 x^{2} d x + \int_{0}^{1} - \frac{10}{3} x d x$

Step 2

Apply the constant rule.

$92 x]_{0}^{1} + \int_{0}^{1} - 2 x^{2} d x + \int_{0}^{1} - \frac{10}{3} x d x$

Step 3

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

$92 x]_{0}^{1} - 2 \int_{0}^{1} x^{2} d x + \int_{0}^{1} - \frac{10}{3} x d x$

Step 4

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

$92 x]_{0}^{1} - 2 (\frac{1}{3} x^{3}]_{0}^{1}) + \int_{0}^{1} - \frac{10}{3} x d x$

Step 5

Combine $\frac{1}{3}$ and $x^{3}$ .

$92 x]_{0}^{1} - 2 (\frac{x^{3}}{3}]_{0}^{1}) + \int_{0}^{1} - \frac{10}{3} x d x$

Step 6

Since $- \frac{10}{3}$ is constant with respect to $x$ , move $- \frac{10}{3}$ out of the integral.

$92 x]_{0}^{1} - 2 (\frac{x^{3}}{3}]_{0}^{1}) - \frac{10}{3} \int_{0}^{1} x d x$

Step 7

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

$92 x]_{0}^{1} - 2 (\frac{x^{3}}{3}]_{0}^{1}) - \frac{10}{3} \frac{1}{2} x^{2}]_{0}^{1}$

Step 8

Substitute and simplify.

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Step 8.1

Evaluate $92 x$ at $1$ and at $0$ .

$(92 \cdot 1) - 92 \cdot 0 - 2 (\frac{x^{3}}{3}]_{0}^{1}) - \frac{10}{3} \frac{1}{2} x^{2}]_{0}^{1}$

Step 8.2

Evaluate $\frac{x^{3}}{3}$ at $1$ and at $0$ .

$92 \cdot 1 - 92 \cdot 0 - 2 (\frac{1^{3}}{3} - \frac{0^{3}}{3}) - \frac{10}{3} \frac{1}{2} x^{2}]_{0}^{1}$

Step 8.3

Evaluate $\frac{1}{2} x^{2}$ at $1$ and at $0$ .

$92 \cdot 1 - 92 \cdot 0 - 2 (\frac{1^{3}}{3} - \frac{0^{3}}{3}) - \frac{10}{3} ((\frac{1}{2} \cdot 1^{2}) - \frac{1}{2} \cdot 0^{2})$

Step 8.4

Simplify.

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Step 8.4.1

Multiply $92$ by $1$ .

$92 - 92 \cdot 0 - 2 (\frac{1^{3}}{3} - \frac{0^{3}}{3}) - \frac{10}{3} ((\frac{1}{2} \cdot 1^{2}) - \frac{1}{2} \cdot 0^{2})$

Step 8.4.2

Multiply $- 92$ by $0$ .

$92 + 0 - 2 (\frac{1^{3}}{3} - \frac{0^{3}}{3}) - \frac{10}{3} ((\frac{1}{2} \cdot 1^{2}) - \frac{1}{2} \cdot 0^{2})$

Step 8.4.3

Add $92$ and $0$ .

$92 - 2 (\frac{1^{3}}{3} - \frac{0^{3}}{3}) - \frac{10}{3} ((\frac{1}{2} \cdot 1^{2}) - \frac{1}{2} \cdot 0^{2})$

Step 8.4.4

One to any power is one.

$92 - 2 (\frac{1}{3} - \frac{0^{3}}{3}) - \frac{10}{3} ((\frac{1}{2} \cdot 1^{2}) - \frac{1}{2} \cdot 0^{2})$

Step 8.4.5

Raising $0$ to any positive power yields $0$ .

$92 - 2 (\frac{1}{3} - \frac{0}{3}) - \frac{10}{3} ((\frac{1}{2} \cdot 1^{2}) - \frac{1}{2} \cdot 0^{2})$

Step 8.4.6

Cancel the common factor of $0$ and $3$ .

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Step 8.4.6.1

Factor $3$ out of $0$ .

$92 - 2 (\frac{1}{3} - \frac{3 (0)}{3}) - \frac{10}{3} ((\frac{1}{2} \cdot 1^{2}) - \frac{1}{2} \cdot 0^{2})$

Step 8.4.6.2

Cancel the common factors.

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Step 8.4.6.2.1

Factor $3$ out of $3$ .

$92 - 2 (\frac{1}{3} - \frac{3 \cdot 0}{3 \cdot 1}) - \frac{10}{3} ((\frac{1}{2} \cdot 1^{2}) - \frac{1}{2} \cdot 0^{2})$

Step 8.4.6.2.2

Cancel the common factor.

$92 - 2 (\frac{1}{3} - \frac{3 \cdot 0}{3 \cdot 1}) - \frac{10}{3} ((\frac{1}{2} \cdot 1^{2}) - \frac{1}{2} \cdot 0^{2})$

Step 8.4.6.2.3

Rewrite the expression.

$92 - 2 (\frac{1}{3} - \frac{0}{1}) - \frac{10}{3} ((\frac{1}{2} \cdot 1^{2}) - \frac{1}{2} \cdot 0^{2})$

Step 8.4.6.2.4

Divide $0$ by $1$ .

$92 - 2 (\frac{1}{3} - 0) - \frac{10}{3} ((\frac{1}{2} \cdot 1^{2}) - \frac{1}{2} \cdot 0^{2})$

Step 8.4.7

Multiply $- 1$ by $0$ .

$92 - 2 (\frac{1}{3} + 0) - \frac{10}{3} ((\frac{1}{2} \cdot 1^{2}) - \frac{1}{2} \cdot 0^{2})$

Step 8.4.8

Add $\frac{1}{3}$ and $0$ .

$92 - 2 (\frac{1}{3}) - \frac{10}{3} ((\frac{1}{2} \cdot 1^{2}) - \frac{1}{2} \cdot 0^{2})$

Step 8.4.9

Combine $- 2$ and $\frac{1}{3}$ .

$92 + \frac{- 2}{3} - \frac{10}{3} ((\frac{1}{2} \cdot 1^{2}) - \frac{1}{2} \cdot 0^{2})$

Step 8.4.10

Move the negative in front of the fraction.

$92 - \frac{2}{3} - \frac{10}{3} ((\frac{1}{2} \cdot 1^{2}) - \frac{1}{2} \cdot 0^{2})$

Step 8.4.11

To write $92$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$92 \cdot \frac{3}{3} - \frac{2}{3} - \frac{10}{3} ((\frac{1}{2} \cdot 1^{2}) - \frac{1}{2} \cdot 0^{2})$

Step 8.4.12

Combine $92$ and $\frac{3}{3}$ .

$\frac{92 \cdot 3}{3} - \frac{2}{3} - \frac{10}{3} ((\frac{1}{2} \cdot 1^{2}) - \frac{1}{2} \cdot 0^{2})$

Step 8.4.13

Combine the numerators over the common denominator.

$\frac{92 \cdot 3 - 2}{3} - \frac{10}{3} ((\frac{1}{2} \cdot 1^{2}) - \frac{1}{2} \cdot 0^{2})$

Step 8.4.14

Simplify the numerator.

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Step 8.4.14.1

Multiply $92$ by $3$ .

$\frac{276 - 2}{3} - \frac{10}{3} ((\frac{1}{2} \cdot 1^{2}) - \frac{1}{2} \cdot 0^{2})$

Step 8.4.14.2

Subtract $2$ from $276$ .

$\frac{274}{3} - \frac{10}{3} ((\frac{1}{2} \cdot 1^{2}) - \frac{1}{2} \cdot 0^{2})$

Step 8.4.15

One to any power is one.

$\frac{274}{3} - \frac{10}{3} (\frac{1}{2} \cdot 1 - \frac{1}{2} \cdot 0^{2})$

Step 8.4.16

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

$\frac{274}{3} - \frac{10}{3} (\frac{1}{2} - \frac{1}{2} \cdot 0^{2})$

Step 8.4.17

Raising $0$ to any positive power yields $0$ .

$\frac{274}{3} - \frac{10}{3} (\frac{1}{2} - \frac{1}{2} \cdot 0)$

Step 8.4.18

Multiply $0$ by $- 1$ .

$\frac{274}{3} - \frac{10}{3} (\frac{1}{2} + 0 (\frac{1}{2}))$

Step 8.4.19

Multiply $0$ by $\frac{1}{2}$ .

$\frac{274}{3} - \frac{10}{3} (\frac{1}{2} + 0)$

Step 8.4.20

Add $\frac{1}{2}$ and $0$ .

$\frac{274}{3} - \frac{10}{3} \cdot \frac{1}{2}$

Step 8.4.21

Multiply $\frac{1}{2}$ by $\frac{10}{3}$ .

$\frac{274}{3} - \frac{10}{2 \cdot 3}$

Step 8.4.22

Multiply $2$ by $3$ .

$\frac{274}{3} - \frac{10}{6}$

Step 8.4.23

Cancel the common factor of $10$ and $6$ .

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Step 8.4.23.1

Factor $2$ out of $10$ .

$\frac{274}{3} - \frac{2 (5)}{6}$

Step 8.4.23.2

Cancel the common factors.

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Step 8.4.23.2.1

Factor $2$ out of $6$ .

$\frac{274}{3} - \frac{2 \cdot 5}{2 \cdot 3}$

Step 8.4.23.2.2

Cancel the common factor.

$\frac{274}{3} - \frac{2 \cdot 5}{2 \cdot 3}$

Step 8.4.23.2.3

Rewrite the expression.

$\frac{274}{3} - \frac{5}{3}$

Step 8.4.24

Combine the numerators over the common denominator.

$\frac{274 - 5}{3}$

Step 8.4.25

Subtract $5$ from $274$ .

$\frac{269}{3}$

Step 9

The result can be shown in multiple forms.

Exact Form:

$\frac{269}{3}$

Decimal Form:

$89.\overline{6}$

Mixed Number Form:

$89 \frac{2}{3}$

Step 10