Calculus Examples

$\int_{0}^{1} (x^{2} - 2 x + 3) d x$

Step 1

Remove parentheses.

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

Step 2

Split the single integral into multiple integrals.

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

Step 3

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

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

Step 4

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

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

Step 5

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

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

Step 6

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

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

Step 7

Apply the constant rule.

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

Step 8

Simplify the answer.

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Step 8.1

Combine $\frac{1}{3} x^{3}]_{0}^{1}$ and $3 x]_{0}^{1}$ .

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

Step 8.2

Substitute and simplify.

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Step 8.2.1

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

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

Step 8.2.2

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

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

Step 8.2.3

Simplify.

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Step 8.2.3.1

One to any power is one.

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

Step 8.2.3.2

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

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

Step 8.2.3.3

Multiply $3$ by $1$ .

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

Step 8.2.3.4

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

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

Step 8.2.3.5

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

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

Step 8.2.3.6

Combine the numerators over the common denominator.

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

Step 8.2.3.7

Simplify the numerator.

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Step 8.2.3.7.1

Multiply $3$ by $3$ .

$\frac{1 + 9}{3} - (\frac{1}{3} \cdot 0^{3} + 3 \cdot 0) - 2 ((\frac{1^{2}}{2}) - \frac{0^{2}}{2})$

Step 8.2.3.7.2

Add $1$ and $9$ .

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

Step 8.2.3.8

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

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

Step 8.2.3.9

Multiply $\frac{1}{3}$ by $0$ .

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

Step 8.2.3.10

Multiply $3$ by $0$ .

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

Step 8.2.3.11

Add $0$ and $0$ .

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

Step 8.2.3.12

Multiply $- 1$ by $0$ .

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

Step 8.2.3.13

Add $\frac{10}{3}$ and $0$ .

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

Step 8.2.3.14

One to any power is one.

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

Step 8.2.3.15

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

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

Step 8.2.3.16

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

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Step 8.2.3.16.1

Factor $2$ out of $0$ .

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

Step 8.2.3.16.2

Cancel the common factors.

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Step 8.2.3.16.2.1

Factor $2$ out of $2$ .

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

Step 8.2.3.16.2.2

Cancel the common factor.

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

Step 8.2.3.16.2.3

Rewrite the expression.

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

Step 8.2.3.16.2.4

Divide $0$ by $1$ .

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

Step 8.2.3.17

Multiply $- 1$ by $0$ .

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

Step 8.2.3.18

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

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

Step 8.2.3.19

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

$\frac{10}{3} + \frac{- 2}{2}$

Step 8.2.3.20

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

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Step 8.2.3.20.1

Factor $2$ out of $- 2$ .

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

Step 8.2.3.20.2

Cancel the common factors.

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Step 8.2.3.20.2.1

Factor $2$ out of $2$ .

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

Step 8.2.3.20.2.2

Cancel the common factor.

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

Step 8.2.3.20.2.3

Rewrite the expression.

$\frac{10}{3} + \frac{- 1}{1}$

Step 8.2.3.20.2.4

Divide $- 1$ by $1$ .

$\frac{10}{3} - 1$

Step 8.2.3.21

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

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

Step 8.2.3.22

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

$\frac{10}{3} + \frac{- 1 \cdot 3}{3}$

Step 8.2.3.23

Combine the numerators over the common denominator.

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

Step 8.2.3.24

Simplify the numerator.

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Step 8.2.3.24.1

Multiply $- 1$ by $3$ .

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

Step 8.2.3.24.2

Subtract $3$ from $10$ .

$\frac{7}{3}$

Step 9

The result can be shown in multiple forms.

Exact Form:

$\frac{7}{3}$

Decimal Form:

$2.\overline{3}$

Mixed Number Form:

$2 \frac{1}{3}$

Step 10