Calculus Examples

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

Since integration is linear, the integral of $2 x^{2} + x$ with respect to $x$ is $\int_{1}^{2} 2 x^{2} d x + \int_{1}^{2} x d x$ .

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

Since $2$ is constant with respect to $x$ , the integral of $2 x^{2}$ with respect to $x$ is $2 \int_{1}^{2} x^{2} d x$ .

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

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

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

Combine fractions.

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Write $x^{3}$ as a fraction with denominator $1$ .

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

Multiply $\frac{1}{3}$ and $\frac{x^{3}}{1}$ to get $\frac{x^{3}}{3}$ .

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

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

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

Simplify the answer.

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Write $x^{2}$ as a fraction with denominator $1$ .

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

Multiply $\frac{1}{2}$ and $\frac{x^{2}}{1}$ to get $\frac{x^{2}}{2}$ .

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

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

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

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

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

Raise $2$ to the power of $3$ to get $8$ .

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

One to any power is one.

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

Combine the numerators over the common denominator.

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

Multiply $- 1$ by $1$ to get $- 1$ .

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

Subtract $1$ from $8$ to get $7$ .

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

Write $2$ as a fraction with denominator $1$ .

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

Multiply $\frac{2}{1}$ and $\frac{7}{3}$ to get $\frac{2 \cdot 7}{3}$ .

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

Multiply $2$ by $7$ to get $14$ .

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

Raise $2$ to the power of $2$ to get $4$ .

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

Reduce the expression by cancelling the common factors.

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Factor $2$ out of $2$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Divide $2$ by $1$ to get $2$ .

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

One to any power is one.

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

To write $\frac{2}{1}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

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

Write each expression with a common denominator of $2$ , by multiplying each by an appropriate factor of $1$ .

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Combine.

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

Multiply $2$ by $1$ to get $2$ .

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

Combine the numerators over the common denominator.

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

Multiply $2$ by $2$ to get $4$ .

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

Multiply $- 1$ by $1$ to get $- 1$ .

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

Subtract $1$ from $4$ to get $3$ .

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

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

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

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

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

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

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Combine.

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

Multiply $3$ by $2$ to get $6$ .

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

Combine.

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

Multiply $2$ by $3$ to get $6$ .

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

Combine the numerators over the common denominator.

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

Multiply $14$ by $2$ to get $28$ .

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

Multiply $3$ by $3$ to get $9$ .

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

Reorder terms.

$\frac{37}{6}$

The result can be shown in both exact and decimal forms.

Exact Form:

$\frac{37}{6}$

Decimal Form:

$6.1\overline{6}$

Mixed Number Form:

$6 \frac{1}{6}$

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Calculus Examples

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