Calculus Examples

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

Step 1

Split the single integral into multiple integrals.

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

Step 2

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

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

Step 3

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

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

Step 4

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

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

Step 5

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

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

Step 6

Substitute and simplify.

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Step 6.1

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

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

Step 6.2

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

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

Step 6.3

Simplify.

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Step 6.3.1

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

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

Step 6.3.2

One to any power is one.

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

Step 6.3.3

Combine the numerators over the common denominator.

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

Step 6.3.4

Subtract $1$ from $8$ .

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

Step 6.3.5

Combine $2$ and $\frac{7}{3}$ .

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

Step 6.3.6

Multiply $2$ by $7$ .

$\frac{14}{3} + \frac{1}{2} \cdot 2^{2} - \frac{1}{2} \cdot 1^{2}$

Step 6.3.7

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

$\frac{14}{3} + \frac{1}{2} \cdot 4 - \frac{1}{2} \cdot 1^{2}$

Step 6.3.8

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

$\frac{14}{3} + \frac{4}{2} - \frac{1}{2} \cdot 1^{2}$

Step 6.3.9

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

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Step 6.3.9.1

Factor $2$ out of $4$ .

$\frac{14}{3} + \frac{2 \cdot 2}{2} - \frac{1}{2} \cdot 1^{2}$

Step 6.3.9.2

Cancel the common factors.

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Step 6.3.9.2.1

Factor $2$ out of $2$ .

$\frac{14}{3} + \frac{2 \cdot 2}{2 (1)} - \frac{1}{2} \cdot 1^{2}$

Step 6.3.9.2.2

Cancel the common factor.

$\frac{14}{3} + \frac{2 \cdot 2}{2 \cdot 1} - \frac{1}{2} \cdot 1^{2}$

Step 6.3.9.2.3

Rewrite the expression.

$\frac{14}{3} + \frac{2}{1} - \frac{1}{2} \cdot 1^{2}$

Step 6.3.9.2.4

Divide $2$ by $1$ .

$\frac{14}{3} + 2 - \frac{1}{2} \cdot 1^{2}$

Step 6.3.10

One to any power is one.

$\frac{14}{3} + 2 - \frac{1}{2} \cdot 1$

Step 6.3.11

Multiply $- 1$ by $1$ .

$\frac{14}{3} + 2 - \frac{1}{2}$

Step 6.3.12

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

$\frac{14}{3} + 2 \cdot \frac{2}{2} - \frac{1}{2}$

Step 6.3.13

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

$\frac{14}{3} + \frac{2 \cdot 2}{2} - \frac{1}{2}$

Step 6.3.14

Combine the numerators over the common denominator.

$\frac{14}{3} + \frac{2 \cdot 2 - 1}{2}$

Step 6.3.15

Simplify the numerator.

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Step 6.3.15.1

Multiply $2$ by $2$ .

$\frac{14}{3} + \frac{4 - 1}{2}$

Step 6.3.15.2

Subtract $1$ from $4$ .

$\frac{14}{3} + \frac{3}{2}$

Step 6.3.16

To write $\frac{14}{3}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{14}{3} \cdot \frac{2}{2} + \frac{3}{2}$

Step 6.3.17

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

$\frac{14}{3} \cdot \frac{2}{2} + \frac{3}{2} \cdot \frac{3}{3}$

Step 6.3.18

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

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Step 6.3.18.1

Multiply $\frac{14}{3}$ by $\frac{2}{2}$ .

$\frac{14 \cdot 2}{3 \cdot 2} + \frac{3}{2} \cdot \frac{3}{3}$

Step 6.3.18.2

Multiply $3$ by $2$ .

$\frac{14 \cdot 2}{6} + \frac{3}{2} \cdot \frac{3}{3}$

Step 6.3.18.3

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

$\frac{14 \cdot 2}{6} + \frac{3 \cdot 3}{2 \cdot 3}$

Step 6.3.18.4

Multiply $2$ by $3$ .

$\frac{14 \cdot 2}{6} + \frac{3 \cdot 3}{6}$

Step 6.3.19

Combine the numerators over the common denominator.

$\frac{14 \cdot 2 + 3 \cdot 3}{6}$

Step 6.3.20

Simplify the numerator.

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Step 6.3.20.1

Multiply $14$ by $2$ .

$\frac{28 + 3 \cdot 3}{6}$

Step 6.3.20.2

Multiply $3$ by $3$ .

$\frac{28 + 9}{6}$

Step 6.3.20.3

Add $28$ and $9$ .

$\frac{37}{6}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$\frac{37}{6}$

Decimal Form:

$6.1\overline{6}$

Mixed Number Form:

$6 \frac{1}{6}$

Step 8

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