Calculus Examples

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

Step 1

Remove parentheses.

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

Step 2

Split the single integral into multiple integrals.

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

Step 3

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

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

Step 4

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

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

Step 5

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

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

Step 6

Simplify the answer.

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Step 6.1

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

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

Step 6.2

Substitute and simplify.

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Step 6.2.1

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

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

Step 6.2.2

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

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

Step 6.2.3

Simplify.

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Step 6.2.3.1

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

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

Step 6.2.3.2

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

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

Step 6.2.3.3

One to any power is one.

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

Step 6.2.3.4

Multiply $- 1$ by $1$ .

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

Step 6.2.3.5

Combine the numerators over the common denominator.

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

Step 6.2.3.6

Subtract $1$ from $8$ .

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

Step 6.2.3.7

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

$\frac{7}{3} - (\frac{4}{2} - \frac{1^{2}}{2})$

Step 6.2.3.8

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

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Step 6.2.3.8.1

Factor $2$ out of $4$ .

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

Step 6.2.3.8.2

Cancel the common factors.

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Step 6.2.3.8.2.1

Factor $2$ out of $2$ .

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

Step 6.2.3.8.2.2

Cancel the common factor.

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

Step 6.2.3.8.2.3

Rewrite the expression.

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

Step 6.2.3.8.2.4

Divide $2$ by $1$ .

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

Step 6.2.3.9

One to any power is one.

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

Step 6.2.3.10

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

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

Step 6.2.3.11

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

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

Step 6.2.3.12

Combine the numerators over the common denominator.

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

Step 6.2.3.13

Simplify the numerator.

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Step 6.2.3.13.1

Multiply $2$ by $2$ .

$\frac{7}{3} - \frac{4 - 1}{2}$

Step 6.2.3.13.2

Subtract $1$ from $4$ .

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

Step 6.2.3.14

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

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

Step 6.2.3.15

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

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

Step 6.2.3.16

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

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Step 6.2.3.16.1

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

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

Step 6.2.3.16.2

Multiply $3$ by $2$ .

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

Step 6.2.3.16.3

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

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

Step 6.2.3.16.4

Multiply $2$ by $3$ .

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

Step 6.2.3.17

Combine the numerators over the common denominator.

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

Step 6.2.3.18

Simplify the numerator.

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Step 6.2.3.18.1

Multiply $7$ by $2$ .

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

Step 6.2.3.18.2

Multiply $- 3$ by $3$ .

$\frac{14 - 9}{6}$

Step 6.2.3.18.3

Subtract $9$ from $14$ .

$\frac{5}{6}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$\frac{5}{6}$

Decimal Form:

$0.8\overline{3}$

Step 8