Calculus Examples

$\int_{0}^{1} 5 x^{2} - 4 x + 8 d x$

Step 1

Split the single integral into multiple integrals.

$\int_{0}^{1} 5 x^{2} d x + \int_{0}^{1} - 4 x d x + \int_{0}^{1} 8 d x$

Step 2

Since $5$ is constant with respect to $x$ , move $5$ out of the integral.

$5 \int_{0}^{1} x^{2} d x + \int_{0}^{1} - 4 x d x + \int_{0}^{1} 8 d x$

Step 3

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

$5 (\frac{1}{3} x^{3}]_{0}^{1}) + \int_{0}^{1} - 4 x d x + \int_{0}^{1} 8 d x$

Step 4

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

$5 (\frac{x^{3}}{3}]_{0}^{1}) + \int_{0}^{1} - 4 x d x + \int_{0}^{1} 8 d x$

Step 5

Since $- 4$ is constant with respect to $x$ , move $- 4$ out of the integral.

$5 (\frac{x^{3}}{3}]_{0}^{1}) - 4 \int_{0}^{1} x d x + \int_{0}^{1} 8 d x$

Step 6

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

$5 (\frac{x^{3}}{3}]_{0}^{1}) - 4 (\frac{1}{2} x^{2}]_{0}^{1}) + \int_{0}^{1} 8 d x$

Step 7

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

$5 (\frac{x^{3}}{3}]_{0}^{1}) - 4 (\frac{x^{2}}{2}]_{0}^{1}) + \int_{0}^{1} 8 d x$

Step 8

Apply the constant rule.

$5 (\frac{x^{3}}{3}]_{0}^{1}) - 4 (\frac{x^{2}}{2}]_{0}^{1}) + 8 x]_{0}^{1}$

Step 9

Substitute and simplify.

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Step 9.1

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

$5 ((\frac{1^{3}}{3}) - \frac{0^{3}}{3}) - 4 (\frac{x^{2}}{2}]_{0}^{1}) + 8 x]_{0}^{1}$

Step 9.2

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

$5 (\frac{1^{3}}{3} - \frac{0^{3}}{3}) - 4 (\frac{1^{2}}{2} - \frac{0^{2}}{2}) + 8 x]_{0}^{1}$

Step 9.3

Evaluate $8 x$ at $1$ and at $0$ .

$5 (\frac{1^{3}}{3} - \frac{0^{3}}{3}) - 4 (\frac{1^{2}}{2} - \frac{0^{2}}{2}) + (8 \cdot 1) - 8 \cdot 0$

Step 9.4

Simplify.

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Step 9.4.1

One to any power is one.

$5 (\frac{1}{3} - \frac{0^{3}}{3}) - 4 (\frac{1^{2}}{2} - \frac{0^{2}}{2}) + (8 \cdot 1) - 8 \cdot 0$

Step 9.4.2

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

$5 (\frac{1}{3} - \frac{0}{3}) - 4 (\frac{1^{2}}{2} - \frac{0^{2}}{2}) + (8 \cdot 1) - 8 \cdot 0$

Step 9.4.3

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

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Step 9.4.3.1

Factor $3$ out of $0$ .

$5 (\frac{1}{3} - \frac{3 (0)}{3}) - 4 (\frac{1^{2}}{2} - \frac{0^{2}}{2}) + (8 \cdot 1) - 8 \cdot 0$

Step 9.4.3.2

Cancel the common factors.

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Step 9.4.3.2.1

Factor $3$ out of $3$ .

$5 (\frac{1}{3} - \frac{3 \cdot 0}{3 \cdot 1}) - 4 (\frac{1^{2}}{2} - \frac{0^{2}}{2}) + (8 \cdot 1) - 8 \cdot 0$

Step 9.4.3.2.2

Cancel the common factor.

$5 (\frac{1}{3} - \frac{3 \cdot 0}{3 \cdot 1}) - 4 (\frac{1^{2}}{2} - \frac{0^{2}}{2}) + (8 \cdot 1) - 8 \cdot 0$

Step 9.4.3.2.3

Rewrite the expression.

$5 (\frac{1}{3} - \frac{0}{1}) - 4 (\frac{1^{2}}{2} - \frac{0^{2}}{2}) + (8 \cdot 1) - 8 \cdot 0$

Step 9.4.3.2.4

Divide $0$ by $1$ .

$5 (\frac{1}{3} - 0) - 4 (\frac{1^{2}}{2} - \frac{0^{2}}{2}) + (8 \cdot 1) - 8 \cdot 0$

Step 9.4.4

Multiply $- 1$ by $0$ .

$5 (\frac{1}{3} + 0) - 4 (\frac{1^{2}}{2} - \frac{0^{2}}{2}) + (8 \cdot 1) - 8 \cdot 0$

Step 9.4.5

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

$5 (\frac{1}{3}) - 4 (\frac{1^{2}}{2} - \frac{0^{2}}{2}) + (8 \cdot 1) - 8 \cdot 0$

Step 9.4.6

Combine $5$ and $\frac{1}{3}$ .

$\frac{5}{3} - 4 (\frac{1^{2}}{2} - \frac{0^{2}}{2}) + (8 \cdot 1) - 8 \cdot 0$

Step 9.4.7

One to any power is one.

$\frac{5}{3} - 4 (\frac{1}{2} - \frac{0^{2}}{2}) + (8 \cdot 1) - 8 \cdot 0$

Step 9.4.8

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

$\frac{5}{3} - 4 (\frac{1}{2} - \frac{0}{2}) + (8 \cdot 1) - 8 \cdot 0$

Step 9.4.9

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

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Step 9.4.9.1

Factor $2$ out of $0$ .

$\frac{5}{3} - 4 (\frac{1}{2} - \frac{2 (0)}{2}) + (8 \cdot 1) - 8 \cdot 0$

Step 9.4.9.2

Cancel the common factors.

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Step 9.4.9.2.1

Factor $2$ out of $2$ .

$\frac{5}{3} - 4 (\frac{1}{2} - \frac{2 \cdot 0}{2 \cdot 1}) + (8 \cdot 1) - 8 \cdot 0$

Step 9.4.9.2.2

Cancel the common factor.

$\frac{5}{3} - 4 (\frac{1}{2} - \frac{2 \cdot 0}{2 \cdot 1}) + (8 \cdot 1) - 8 \cdot 0$

Step 9.4.9.2.3

Rewrite the expression.

$\frac{5}{3} - 4 (\frac{1}{2} - \frac{0}{1}) + (8 \cdot 1) - 8 \cdot 0$

Step 9.4.9.2.4

Divide $0$ by $1$ .

$\frac{5}{3} - 4 (\frac{1}{2} - 0) + (8 \cdot 1) - 8 \cdot 0$

Step 9.4.10

Multiply $- 1$ by $0$ .

$\frac{5}{3} - 4 (\frac{1}{2} + 0) + (8 \cdot 1) - 8 \cdot 0$

Step 9.4.11

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

$\frac{5}{3} - 4 (\frac{1}{2}) + (8 \cdot 1) - 8 \cdot 0$

Step 9.4.12

Combine $- 4$ and $\frac{1}{2}$ .

$\frac{5}{3} + \frac{- 4}{2} + (8 \cdot 1) - 8 \cdot 0$

Step 9.4.13

Cancel the common factor of $- 4$ and $2$ .

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Step 9.4.13.1

Factor $2$ out of $- 4$ .

$\frac{5}{3} + \frac{2 \cdot - 2}{2} + (8 \cdot 1) - 8 \cdot 0$

Step 9.4.13.2

Cancel the common factors.

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Step 9.4.13.2.1

Factor $2$ out of $2$ .

$\frac{5}{3} + \frac{2 \cdot - 2}{2 (1)} + (8 \cdot 1) - 8 \cdot 0$

Step 9.4.13.2.2

Cancel the common factor.

$\frac{5}{3} + \frac{2 \cdot - 2}{2 \cdot 1} + (8 \cdot 1) - 8 \cdot 0$

Step 9.4.13.2.3

Rewrite the expression.

$\frac{5}{3} + \frac{- 2}{1} + (8 \cdot 1) - 8 \cdot 0$

Step 9.4.13.2.4

Divide $- 2$ by $1$ .

$\frac{5}{3} - 2 + (8 \cdot 1) - 8 \cdot 0$

Step 9.4.14

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

$\frac{5}{3} - 2 \cdot \frac{3}{3} + (8 \cdot 1) - 8 \cdot 0$

Step 9.4.15

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

$\frac{5}{3} + \frac{- 2 \cdot 3}{3} + (8 \cdot 1) - 8 \cdot 0$

Step 9.4.16

Combine the numerators over the common denominator.

$\frac{5 - 2 \cdot 3}{3} + (8 \cdot 1) - 8 \cdot 0$

Step 9.4.17

Simplify the numerator.

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Step 9.4.17.1

Multiply $- 2$ by $3$ .

$\frac{5 - 6}{3} + (8 \cdot 1) - 8 \cdot 0$

Step 9.4.17.2

Subtract $6$ from $5$ .

$\frac{- 1}{3} + (8 \cdot 1) - 8 \cdot 0$

Step 9.4.18

Move the negative in front of the fraction.

$- \frac{1}{3} + (8 \cdot 1) - 8 \cdot 0$

Step 9.4.19

Multiply $8$ by $1$ .

$- \frac{1}{3} + 8 - 8 \cdot 0$

Step 9.4.20

Multiply $- 8$ by $0$ .

$- \frac{1}{3} + 8 + 0$

Step 9.4.21

Add $8$ and $0$ .

$- \frac{1}{3} + 8$

Step 9.4.22

To write $8$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$- \frac{1}{3} + 8 \cdot \frac{3}{3}$

Step 9.4.23

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

$- \frac{1}{3} + \frac{8 \cdot 3}{3}$

Step 9.4.24

Combine the numerators over the common denominator.

$\frac{- 1 + 8 \cdot 3}{3}$

Step 9.4.25

Simplify the numerator.

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Step 9.4.25.1

Multiply $8$ by $3$ .

$\frac{- 1 + 24}{3}$

Step 9.4.25.2

Add $- 1$ and $24$ .

$\frac{23}{3}$

Step 10

The result can be shown in multiple forms.

Exact Form:

$\frac{23}{3}$

Decimal Form:

$7.\overline{6}$

Mixed Number Form:

$7 \frac{2}{3}$

Step 11