Calculus Examples

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

Step 1

Simplify.

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Step 1.1

Apply the distributive property.

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

Step 1.2

Multiply $3$ by $2$ .

$\int_{1}^{2} (2 x^{- 2} + 6 x) x^{2} d x$

Step 1.3

Simplify each term.

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Step 1.3.1

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\int_{1}^{2} (2 \frac{1}{x^{2}} + 6 x) x^{2} d x$

Step 1.3.2

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

$\int_{1}^{2} (\frac{2}{x^{2}} + 6 x) x^{2} d x$

Step 1.4

Apply the distributive property.

$\int_{1}^{2} \frac{2}{x^{2}} x^{2} + 6 x \cdot x^{2} d x$

Step 1.5

Cancel the common factor of $x^{2}$ .

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Step 1.5.1

Cancel the common factor.

$\int_{1}^{2} \frac{2}{x^{2}} x^{2} + 6 x \cdot x^{2} d x$

Step 1.5.2

Rewrite the expression.

$\int_{1}^{2} 2 + 6 x \cdot x^{2} d x$

Step 1.6

Multiply $x$ by $x^{2}$ by adding the exponents.

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Step 1.6.1

Move $x^{2}$ .

$\int_{1}^{2} 2 + 6 (x^{2} x) d x$

Step 1.6.2

Multiply $x^{2}$ by $x$ .

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Step 1.6.2.1

Raise $x$ to the power of $1$ .

$\int_{1}^{2} 2 + 6 (x^{2} x^{1}) d x$

Step 1.6.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\int_{1}^{2} 2 + 6 x^{2 + 1} d x$

Step 1.6.3

Add $2$ and $1$ .

$\int_{1}^{2} 2 + 6 x^{3} d x$

Step 2

Split the single integral into multiple integrals.

$\int_{1}^{2} 2 d x + \int_{1}^{2} 6 x^{3} d x$

Step 3

Apply the constant rule.

$2 x]_{1}^{2} + \int_{1}^{2} 6 x^{3} d x$

Step 4

Since $6$ is constant with respect to $x$ , move $6$ out of the integral.

$2 x]_{1}^{2} + 6 \int_{1}^{2} x^{3} d x$

Step 5

By the Power Rule, the integral of $x^{3}$ with respect to $x$ is $\frac{1}{4} x^{4}$ .

$2 x]_{1}^{2} + 6 (\frac{1}{4} x^{4}]_{1}^{2})$

Step 6

Simplify the answer.

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Step 6.1

Combine $\frac{1}{4}$ and $x^{4}$ .

$2 x]_{1}^{2} + 6 (\frac{x^{4}}{4}]_{1}^{2})$

Step 6.2

Substitute and simplify.

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Step 6.2.1

Evaluate $2 x$ at $2$ and at $1$ .

$(2 \cdot 2) - 2 \cdot 1 + 6 (\frac{x^{4}}{4}]_{1}^{2})$

Step 6.2.2

Evaluate $\frac{x^{4}}{4}$ at $2$ and at $1$ .

$2 \cdot 2 - 2 \cdot 1 + 6 (\frac{2^{4}}{4} - \frac{1^{4}}{4})$

Step 6.2.3

Simplify.

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Step 6.2.3.1

Multiply $2$ by $2$ .

$4 - 2 \cdot 1 + 6 (\frac{2^{4}}{4} - \frac{1^{4}}{4})$

Step 6.2.3.2

Multiply $- 2$ by $1$ .

$4 - 2 + 6 (\frac{2^{4}}{4} - \frac{1^{4}}{4})$

Step 6.2.3.3

Subtract $2$ from $4$ .

$2 + 6 (\frac{2^{4}}{4} - \frac{1^{4}}{4})$

Step 6.2.3.4

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

$2 + 6 (\frac{16}{4} - \frac{1^{4}}{4})$

Step 6.2.3.5

Cancel the common factor of $16$ and $4$ .

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Step 6.2.3.5.1

Factor $4$ out of $16$ .

$2 + 6 (\frac{4 \cdot 4}{4} - \frac{1^{4}}{4})$

Step 6.2.3.5.2

Cancel the common factors.

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Step 6.2.3.5.2.1

Factor $4$ out of $4$ .

$2 + 6 (\frac{4 \cdot 4}{4 (1)} - \frac{1^{4}}{4})$

Step 6.2.3.5.2.2

Cancel the common factor.

$2 + 6 (\frac{4 \cdot 4}{4 \cdot 1} - \frac{1^{4}}{4})$

Step 6.2.3.5.2.3

Rewrite the expression.

$2 + 6 (\frac{4}{1} - \frac{1^{4}}{4})$

Step 6.2.3.5.2.4

Divide $4$ by $1$ .

$2 + 6 (4 - \frac{1^{4}}{4})$

Step 6.2.3.6

One to any power is one.

$2 + 6 (4 - \frac{1}{4})$

Step 6.2.3.7

To write $4$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$2 + 6 (4 \cdot \frac{4}{4} - \frac{1}{4})$

Step 6.2.3.8

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

$2 + 6 (\frac{4 \cdot 4}{4} - \frac{1}{4})$

Step 6.2.3.9

Combine the numerators over the common denominator.

$2 + 6 \frac{4 \cdot 4 - 1}{4}$

Step 6.2.3.10

Simplify the numerator.

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Step 6.2.3.10.1

Multiply $4$ by $4$ .

$2 + 6 \frac{16 - 1}{4}$

Step 6.2.3.10.2

Subtract $1$ from $16$ .

$2 + 6 (\frac{15}{4})$

Step 6.2.3.11

Combine $6$ and $\frac{15}{4}$ .

$2 + \frac{6 \cdot 15}{4}$

Step 6.2.3.12

Multiply $6$ by $15$ .

$2 + \frac{90}{4}$

Step 6.2.3.13

Cancel the common factor of $90$ and $4$ .

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Step 6.2.3.13.1

Factor $2$ out of $90$ .

$2 + \frac{2 (45)}{4}$

Step 6.2.3.13.2

Cancel the common factors.

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Step 6.2.3.13.2.1

Factor $2$ out of $4$ .

$2 + \frac{2 \cdot 45}{2 \cdot 2}$

Step 6.2.3.13.2.2

Cancel the common factor.

$2 + \frac{2 \cdot 45}{2 \cdot 2}$

Step 6.2.3.13.2.3

Rewrite the expression.

$2 + \frac{45}{2}$

Step 6.2.3.14

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

$2 \cdot \frac{2}{2} + \frac{45}{2}$

Step 6.2.3.15

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

$\frac{2 \cdot 2}{2} + \frac{45}{2}$

Step 6.2.3.16

Combine the numerators over the common denominator.

$\frac{2 \cdot 2 + 45}{2}$

Step 6.2.3.17

Simplify the numerator.

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Step 6.2.3.17.1

Multiply $2$ by $2$ .

$\frac{4 + 45}{2}$

Step 6.2.3.17.2

Add $4$ and $45$ .

$\frac{49}{2}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$\frac{49}{2}$

Decimal Form:

$24.5$

Mixed Number Form:

$24 \frac{1}{2}$

Step 8