Calculus Examples

$\int_{2}^{5} 3 x - \frac{6}{7 x^{2}} d x$

Step 1

Split the single integral into multiple integrals.

$\int_{2}^{5} 3 x d x + \int_{2}^{5} - \frac{6}{7 x^{2}} d x$

Step 2

Since $3$ is constant with respect to $x$ , move $3$ out of the integral.

$3 \int_{2}^{5} x d x + \int_{2}^{5} - \frac{6}{7 x^{2}} d x$

Step 3

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

$3 (\frac{1}{2} x^{2}]_{2}^{5}) + \int_{2}^{5} - \frac{6}{7 x^{2}} d x$

Step 4

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

$3 (\frac{x^{2}}{2}]_{2}^{5}) + \int_{2}^{5} - \frac{6}{7 x^{2}} d x$

Step 5

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

$3 (\frac{x^{2}}{2}]_{2}^{5}) - \int_{2}^{5} \frac{6}{7 x^{2}} d x$

Step 6

Since $\frac{6}{7}$ is constant with respect to $x$ , move $\frac{6}{7}$ out of the integral.

$3 (\frac{x^{2}}{2}]_{2}^{5}) - (\frac{6}{7} \int_{2}^{5} \frac{1}{x^{2}} d x)$

Step 7

Apply basic rules of exponents.

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Step 7.1

Move $x^{2}$ out of the denominator by raising it to the $- 1$ power.

$3 (\frac{x^{2}}{2}]_{2}^{5}) - \frac{6}{7} \int_{2}^{5} {(x^{2})}^{- 1} d x$

Step 7.2

Multiply the exponents in ${(x^{2})}^{- 1}$ .

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Step 7.2.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$3 (\frac{x^{2}}{2}]_{2}^{5}) - \frac{6}{7} \int_{2}^{5} x^{2 \cdot - 1} d x$

Step 7.2.2

Multiply $2$ by $- 1$ .

$3 (\frac{x^{2}}{2}]_{2}^{5}) - \frac{6}{7} \int_{2}^{5} x^{- 2} d x$

Step 8

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

$3 (\frac{x^{2}}{2}]_{2}^{5}) - \frac{6}{7} - x^{- 1}]_{2}^{5}$

Step 9

Substitute and simplify.

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Step 9.1

Evaluate $\frac{x^{2}}{2}$ at $5$ and at $2$ .

$3 ((\frac{5^{2}}{2}) - \frac{2^{2}}{2}) - \frac{6}{7} - x^{- 1}]_{2}^{5}$

Step 9.2

Evaluate $- x^{- 1}$ at $5$ and at $2$ .

$3 (\frac{5^{2}}{2} - \frac{2^{2}}{2}) - \frac{6}{7} (- 5^{- 1} + 2^{- 1})$

Step 9.3

Simplify.

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Step 9.3.1

Raise $5$ to the power of $2$ .

$3 (\frac{25}{2} - \frac{2^{2}}{2}) - \frac{6}{7} (- 5^{- 1} + 2^{- 1})$

Step 9.3.2

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

$3 (\frac{25}{2} - \frac{4}{2}) - \frac{6}{7} (- 5^{- 1} + 2^{- 1})$

Step 9.3.3

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

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Step 9.3.3.1

Factor $2$ out of $4$ .

$3 (\frac{25}{2} - \frac{2 \cdot 2}{2}) - \frac{6}{7} (- 5^{- 1} + 2^{- 1})$

Step 9.3.3.2

Cancel the common factors.

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Step 9.3.3.2.1

Factor $2$ out of $2$ .

$3 (\frac{25}{2} - \frac{2 \cdot 2}{2 (1)}) - \frac{6}{7} (- 5^{- 1} + 2^{- 1})$

Step 9.3.3.2.2

Cancel the common factor.

$3 (\frac{25}{2} - \frac{2 \cdot 2}{2 \cdot 1}) - \frac{6}{7} (- 5^{- 1} + 2^{- 1})$

Step 9.3.3.2.3

Rewrite the expression.

$3 (\frac{25}{2} - \frac{2}{1}) - \frac{6}{7} (- 5^{- 1} + 2^{- 1})$

Step 9.3.3.2.4

Divide $2$ by $1$ .

$3 (\frac{25}{2} - 1 \cdot 2) - \frac{6}{7} (- 5^{- 1} + 2^{- 1})$

Step 9.3.4

Multiply $- 1$ by $2$ .

$3 (\frac{25}{2} - 2) - \frac{6}{7} (- 5^{- 1} + 2^{- 1})$

Step 9.3.5

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

$3 (\frac{25}{2} - 2 \cdot \frac{2}{2}) - \frac{6}{7} (- 5^{- 1} + 2^{- 1})$

Step 9.3.6

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

$3 (\frac{25}{2} + \frac{- 2 \cdot 2}{2}) - \frac{6}{7} (- 5^{- 1} + 2^{- 1})$

Step 9.3.7

Combine the numerators over the common denominator.

$3 \frac{25 - 2 \cdot 2}{2} - \frac{6}{7} (- 5^{- 1} + 2^{- 1})$

Step 9.3.8

Simplify the numerator.

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Step 9.3.8.1

Multiply $- 2$ by $2$ .

$3 \frac{25 - 4}{2} - \frac{6}{7} (- 5^{- 1} + 2^{- 1})$

Step 9.3.8.2

Subtract $4$ from $25$ .

$3 (\frac{21}{2}) - \frac{6}{7} (- 5^{- 1} + 2^{- 1})$

Step 9.3.9

Combine $3$ and $\frac{21}{2}$ .

$\frac{3 \cdot 21}{2} - \frac{6}{7} (- 5^{- 1} + 2^{- 1})$

Step 9.3.10

Multiply $3$ by $21$ .

$\frac{63}{2} - \frac{6}{7} (- 5^{- 1} + 2^{- 1})$

Step 9.3.11

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

$\frac{63}{2} - \frac{6}{7} (- \frac{1}{5} + 2^{- 1})$

Step 9.3.12

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

$\frac{63}{2} - \frac{6}{7} (- \frac{1}{5} + \frac{1}{2})$

Step 9.3.13

To write $- \frac{1}{5}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{63}{2} - \frac{6}{7} (- \frac{1}{5} \cdot \frac{2}{2} + \frac{1}{2})$

Step 9.3.14

To write $\frac{1}{2}$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$\frac{63}{2} - \frac{6}{7} (- \frac{1}{5} \cdot \frac{2}{2} + \frac{1}{2} \cdot \frac{5}{5})$

Step 9.3.15

Write each expression with a common denominator of $10$ , by multiplying each by an appropriate factor of $1$ .

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Step 9.3.15.1

Multiply $\frac{1}{5}$ by $\frac{2}{2}$ .

$\frac{63}{2} - \frac{6}{7} (- \frac{2}{5 \cdot 2} + \frac{1}{2} \cdot \frac{5}{5})$

Step 9.3.15.2

Multiply $5$ by $2$ .

$\frac{63}{2} - \frac{6}{7} (- \frac{2}{10} + \frac{1}{2} \cdot \frac{5}{5})$

Step 9.3.15.3

Multiply $\frac{1}{2}$ by $\frac{5}{5}$ .

$\frac{63}{2} - \frac{6}{7} (- \frac{2}{10} + \frac{5}{2 \cdot 5})$

Step 9.3.15.4

Multiply $2$ by $5$ .

$\frac{63}{2} - \frac{6}{7} (- \frac{2}{10} + \frac{5}{10})$

Step 9.3.16

Combine the numerators over the common denominator.

$\frac{63}{2} - \frac{6}{7} \cdot \frac{- 2 + 5}{10}$

Step 9.3.17

Add $- 2$ and $5$ .

$\frac{63}{2} - \frac{6}{7} \cdot \frac{3}{10}$

Step 9.3.18

Multiply $\frac{3}{10}$ by $\frac{6}{7}$ .

$\frac{63}{2} - \frac{3 \cdot 6}{10 \cdot 7}$

Step 9.3.19

Multiply $3$ by $6$ .

$\frac{63}{2} - \frac{18}{10 \cdot 7}$

Step 9.3.20

Multiply $10$ by $7$ .

$\frac{63}{2} - \frac{18}{70}$

Step 9.3.21

Cancel the common factor of $18$ and $70$ .

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Step 9.3.21.1

Factor $2$ out of $18$ .

$\frac{63}{2} - \frac{2 (9)}{70}$

Step 9.3.21.2

Cancel the common factors.

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Step 9.3.21.2.1

Factor $2$ out of $70$ .

$\frac{63}{2} - \frac{2 \cdot 9}{2 \cdot 35}$

Step 9.3.21.2.2

Cancel the common factor.

$\frac{63}{2} - \frac{2 \cdot 9}{2 \cdot 35}$

Step 9.3.21.2.3

Rewrite the expression.

$\frac{63}{2} - \frac{9}{35}$

Step 9.3.22

To write $\frac{63}{2}$ as a fraction with a common denominator, multiply by $\frac{35}{35}$ .

$\frac{63}{2} \cdot \frac{35}{35} - \frac{9}{35}$

Step 9.3.23

To write $- \frac{9}{35}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{63}{2} \cdot \frac{35}{35} - \frac{9}{35} \cdot \frac{2}{2}$

Step 9.3.24

Write each expression with a common denominator of $70$ , by multiplying each by an appropriate factor of $1$ .

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Step 9.3.24.1

Multiply $\frac{63}{2}$ by $\frac{35}{35}$ .

$\frac{63 \cdot 35}{2 \cdot 35} - \frac{9}{35} \cdot \frac{2}{2}$

Step 9.3.24.2

Multiply $2$ by $35$ .

$\frac{63 \cdot 35}{70} - \frac{9}{35} \cdot \frac{2}{2}$

Step 9.3.24.3

Multiply $\frac{9}{35}$ by $\frac{2}{2}$ .

$\frac{63 \cdot 35}{70} - \frac{9 \cdot 2}{35 \cdot 2}$

Step 9.3.24.4

Multiply $35$ by $2$ .

$\frac{63 \cdot 35}{70} - \frac{9 \cdot 2}{70}$

Step 9.3.25

Combine the numerators over the common denominator.

$\frac{63 \cdot 35 - 9 \cdot 2}{70}$

Step 9.3.26

Simplify the numerator.

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Step 9.3.26.1

Multiply $63$ by $35$ .

$\frac{2205 - 9 \cdot 2}{70}$

Step 9.3.26.2

Multiply $- 9$ by $2$ .

$\frac{2205 - 18}{70}$

Step 9.3.26.3

Subtract $18$ from $2205$ .

$\frac{2187}{70}$

Step 10

The result can be shown in multiple forms.

Exact Form:

$\frac{2187}{70}$

Decimal Form:

$31.24285714 \dots$

Mixed Number Form:

$31 \frac{17}{70}$

Step 11