Calculus Examples

$\int_{4}^{9} \frac{x - \sqrt{x}}{x^{3}} d x$

Step 1

Move $x^{3}$ out of the denominator by raising it to the $- 1$ power.

$\int_{4}^{9} (x - \sqrt{x}) {(x^{3})}^{- 1} d x$

Step 2

Multiply the exponents in ${(x^{3})}^{- 1}$ .

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Step 2.1

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

$\int_{4}^{9} (x - \sqrt{x}) x^{3 \cdot - 1} d x$

Step 2.2

Multiply $3$ by $- 1$ .

$\int_{4}^{9} (x - \sqrt{x}) x^{- 3} d x$

Step 3

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x}$ as $x^{\frac{1}{2}}$ .

$\int_{4}^{9} (x - x^{\frac{1}{2}}) x^{- 3} d x$

Step 4

Simplify.

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Step 4.1

Apply the distributive property.

$\int_{4}^{9} x \cdot x^{- 3} - x^{\frac{1}{2}} x^{- 3} d x$

Step 4.2

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

$\int_{4}^{9} x^{1} x^{- 3} - x^{\frac{1}{2}} x^{- 3} d x$

Step 4.3

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

$\int_{4}^{9} x^{1 - 3} - x^{\frac{1}{2}} x^{- 3} d x$

Step 4.4

Subtract $3$ from $1$ .

$\int_{4}^{9} x^{- 2} - x^{\frac{1}{2}} x^{- 3} d x$

Step 4.5

Factor out negative.

$\int_{4}^{9} x^{- 2} - (x^{\frac{1}{2}} x^{- 3}) d x$

Step 4.6

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

$\int_{4}^{9} x^{- 2} - x^{\frac{1}{2} - 3} d x$

Step 4.7

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

$\int_{4}^{9} x^{- 2} - x^{\frac{1}{2} - 3 \cdot \frac{2}{2}} d x$

Step 4.8

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

$\int_{4}^{9} x^{- 2} - x^{\frac{1}{2} + \frac{- 3 \cdot 2}{2}} d x$

Step 4.9

Combine the numerators over the common denominator.

$\int_{4}^{9} x^{- 2} - x^{\frac{1 - 3 \cdot 2}{2}} d x$

Step 4.10

Simplify the numerator.

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Step 4.10.1

Multiply $- 3$ by $2$ .

$\int_{4}^{9} x^{- 2} - x^{\frac{1 - 6}{2}} d x$

Step 4.10.2

Subtract $6$ from $1$ .

$\int_{4}^{9} x^{- 2} - x^{\frac{- 5}{2}} d x$

Step 5

Move the negative in front of the fraction.

$\int_{4}^{9} x^{- 2} - x^{- \frac{5}{2}} d x$

Step 6

Split the single integral into multiple integrals.

$\int_{4}^{9} x^{- 2} d x + \int_{4}^{9} - x^{- \frac{5}{2}} d x$

Step 7

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

$- x^{- 1}]_{4}^{9} + \int_{4}^{9} - x^{- \frac{5}{2}} d x$

Step 8

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

$- x^{- 1}]_{4}^{9} - \int_{4}^{9} x^{- \frac{5}{2}} d x$

Step 9

By the Power Rule, the integral of $x^{- \frac{5}{2}}$ with respect to $x$ is $- \frac{2}{3} x^{- \frac{3}{2}}$ .

$- x^{- 1}]_{4}^{9} - (- \frac{2}{3} x^{- \frac{3}{2}}]_{4}^{9})$

Step 10

Simplify the answer.

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Step 10.1

Simplify.

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Step 10.1.1

Combine $x^{- \frac{3}{2}}$ and $\frac{2}{3}$ .

$- x^{- 1}]_{4}^{9} - (- \frac{x^{- \frac{3}{2}} \cdot 2}{3}]_{4}^{9})$

Step 10.1.2

Move $2$ to the left of $x^{- \frac{3}{2}}$ .

$- x^{- 1}]_{4}^{9} - (- \frac{2 \cdot x^{- \frac{3}{2}}}{3}]_{4}^{9})$

Step 10.1.3

Move $x^{- \frac{3}{2}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$- x^{- 1}]_{4}^{9} - (- \frac{2}{3 x^{\frac{3}{2}}}]_{4}^{9})$

Step 10.2

Substitute and simplify.

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Step 10.2.1

Evaluate $- x^{- 1}$ at $9$ and at $4$ .

$(- 9^{- 1}) + 4^{- 1} - (- \frac{2}{3 x^{\frac{3}{2}}}]_{4}^{9})$

Step 10.2.2

Evaluate $- \frac{2}{3 x^{\frac{3}{2}}}$ at $9$ and at $4$ .

$(- 9^{- 1}) + 4^{- 1} - ((- \frac{2}{3 \cdot 9^{\frac{3}{2}}}) + \frac{2}{3 \cdot 4^{\frac{3}{2}}})$

Step 10.2.3

Simplify.

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Step 10.2.3.1

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

$- \frac{1}{9} + 4^{- 1} - ((- \frac{2}{3 \cdot 9^{\frac{3}{2}}}) + \frac{2}{3 \cdot 4^{\frac{3}{2}}})$

Step 10.2.3.2

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

$- \frac{1}{9} + \frac{1}{4} - ((- \frac{2}{3 \cdot 9^{\frac{3}{2}}}) + \frac{2}{3 \cdot 4^{\frac{3}{2}}})$

Step 10.2.3.3

To write $- \frac{1}{9}$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$- \frac{1}{9} \cdot \frac{4}{4} + \frac{1}{4} - ((- \frac{2}{3 \cdot 9^{\frac{3}{2}}}) + \frac{2}{3 \cdot 4^{\frac{3}{2}}})$

Step 10.2.3.4

To write $\frac{1}{4}$ as a fraction with a common denominator, multiply by $\frac{9}{9}$ .

$- \frac{1}{9} \cdot \frac{4}{4} + \frac{1}{4} \cdot \frac{9}{9} - ((- \frac{2}{3 \cdot 9^{\frac{3}{2}}}) + \frac{2}{3 \cdot 4^{\frac{3}{2}}})$

Step 10.2.3.5

Write each expression with a common denominator of $36$ , by multiplying each by an appropriate factor of $1$ .

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Step 10.2.3.5.1

Multiply $\frac{1}{9}$ by $\frac{4}{4}$ .

$- \frac{4}{9 \cdot 4} + \frac{1}{4} \cdot \frac{9}{9} - ((- \frac{2}{3 \cdot 9^{\frac{3}{2}}}) + \frac{2}{3 \cdot 4^{\frac{3}{2}}})$

Step 10.2.3.5.2

Multiply $9$ by $4$ .

$- \frac{4}{36} + \frac{1}{4} \cdot \frac{9}{9} - ((- \frac{2}{3 \cdot 9^{\frac{3}{2}}}) + \frac{2}{3 \cdot 4^{\frac{3}{2}}})$

Step 10.2.3.5.3

Multiply $\frac{1}{4}$ by $\frac{9}{9}$ .

$- \frac{4}{36} + \frac{9}{4 \cdot 9} - ((- \frac{2}{3 \cdot 9^{\frac{3}{2}}}) + \frac{2}{3 \cdot 4^{\frac{3}{2}}})$

Step 10.2.3.5.4

Multiply $4$ by $9$ .

$- \frac{4}{36} + \frac{9}{36} - ((- \frac{2}{3 \cdot 9^{\frac{3}{2}}}) + \frac{2}{3 \cdot 4^{\frac{3}{2}}})$

Step 10.2.3.6

Combine the numerators over the common denominator.

$\frac{- 4 + 9}{36} - ((- \frac{2}{3 \cdot 9^{\frac{3}{2}}}) + \frac{2}{3 \cdot 4^{\frac{3}{2}}})$

Step 10.2.3.7

Add $- 4$ and $9$ .

$\frac{5}{36} - ((- \frac{2}{3 \cdot 9^{\frac{3}{2}}}) + \frac{2}{3 \cdot 4^{\frac{3}{2}}})$

Step 10.2.3.8

Rewrite $9$ as $3^{2}$ .

$\frac{5}{36} - (- \frac{2}{3 \cdot {(3^{2})}^{\frac{3}{2}}} + \frac{2}{3 \cdot 4^{\frac{3}{2}}})$

Step 10.2.3.9

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

$\frac{5}{36} - (- \frac{2}{3 \cdot 3^{2 (\frac{3}{2})}} + \frac{2}{3 \cdot 4^{\frac{3}{2}}})$

Step 10.2.3.10

Cancel the common factor of $2$ .

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Step 10.2.3.10.1

Cancel the common factor.

$\frac{5}{36} - (- \frac{2}{3 \cdot 3^{2 (\frac{3}{2})}} + \frac{2}{3 \cdot 4^{\frac{3}{2}}})$

Step 10.2.3.10.2

Rewrite the expression.

$\frac{5}{36} - (- \frac{2}{3 \cdot 3^{3}} + \frac{2}{3 \cdot 4^{\frac{3}{2}}})$

Step 10.2.3.11

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

$\frac{5}{36} - (- \frac{2}{3 \cdot 27} + \frac{2}{3 \cdot 4^{\frac{3}{2}}})$

Step 10.2.3.12

Multiply $3$ by $27$ .

$\frac{5}{36} - (- \frac{2}{81} + \frac{2}{3 \cdot 4^{\frac{3}{2}}})$

Step 10.2.3.13

Rewrite $4$ as $2^{2}$ .

$\frac{5}{36} - (- \frac{2}{81} + \frac{2}{3 \cdot {(2^{2})}^{\frac{3}{2}}})$

Step 10.2.3.14

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

$\frac{5}{36} - (- \frac{2}{81} + \frac{2}{3 \cdot 2^{2 (\frac{3}{2})}})$

Step 10.2.3.15

Cancel the common factor of $2$ .

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Step 10.2.3.15.1

Cancel the common factor.

$\frac{5}{36} - (- \frac{2}{81} + \frac{2}{3 \cdot 2^{2 (\frac{3}{2})}})$

Step 10.2.3.15.2

Rewrite the expression.

$\frac{5}{36} - (- \frac{2}{81} + \frac{2}{3 \cdot 2^{3}})$

Step 10.2.3.16

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

$\frac{5}{36} - (- \frac{2}{81} + \frac{2}{3 \cdot 8})$

Step 10.2.3.17

Multiply $3$ by $8$ .

$\frac{5}{36} - (- \frac{2}{81} + \frac{2}{24})$

Step 10.2.3.18

Cancel the common factor of $2$ and $24$ .

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Step 10.2.3.18.1

Factor $2$ out of $2$ .

$\frac{5}{36} - (- \frac{2}{81} + \frac{2 (1)}{24})$

Step 10.2.3.18.2

Cancel the common factors.

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Step 10.2.3.18.2.1

Factor $2$ out of $24$ .

$\frac{5}{36} - (- \frac{2}{81} + \frac{2 \cdot 1}{2 \cdot 12})$

Step 10.2.3.18.2.2

Cancel the common factor.

$\frac{5}{36} - (- \frac{2}{81} + \frac{2 \cdot 1}{2 \cdot 12})$

Step 10.2.3.18.2.3

Rewrite the expression.

$\frac{5}{36} - (- \frac{2}{81} + \frac{1}{12})$

Step 10.2.3.19

To write $- \frac{2}{81}$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$\frac{5}{36} - (- \frac{2}{81} \cdot \frac{4}{4} + \frac{1}{12})$

Step 10.2.3.20

To write $\frac{1}{12}$ as a fraction with a common denominator, multiply by $\frac{27}{27}$ .

$\frac{5}{36} - (- \frac{2}{81} \cdot \frac{4}{4} + \frac{1}{12} \cdot \frac{27}{27})$

Step 10.2.3.21

Write each expression with a common denominator of $324$ , by multiplying each by an appropriate factor of $1$ .

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Step 10.2.3.21.1

Multiply $\frac{2}{81}$ by $\frac{4}{4}$ .

$\frac{5}{36} - (- \frac{2 \cdot 4}{81 \cdot 4} + \frac{1}{12} \cdot \frac{27}{27})$

Step 10.2.3.21.2

Multiply $81$ by $4$ .

$\frac{5}{36} - (- \frac{2 \cdot 4}{324} + \frac{1}{12} \cdot \frac{27}{27})$

Step 10.2.3.21.3

Multiply $\frac{1}{12}$ by $\frac{27}{27}$ .

$\frac{5}{36} - (- \frac{2 \cdot 4}{324} + \frac{27}{12 \cdot 27})$

Step 10.2.3.21.4

Multiply $12$ by $27$ .

$\frac{5}{36} - (- \frac{2 \cdot 4}{324} + \frac{27}{324})$

Step 10.2.3.22

Combine the numerators over the common denominator.

$\frac{5}{36} - \frac{- 2 \cdot 4 + 27}{324}$

Step 10.2.3.23

Simplify the numerator.

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Step 10.2.3.23.1

Multiply $- 2$ by $4$ .

$\frac{5}{36} - \frac{- 8 + 27}{324}$

Step 10.2.3.23.2

Add $- 8$ and $27$ .

$\frac{5}{36} - \frac{19}{324}$

Step 10.2.3.24

To write $\frac{5}{36}$ as a fraction with a common denominator, multiply by $\frac{9}{9}$ .

$\frac{5}{36} \cdot \frac{9}{9} - \frac{19}{324}$

Step 10.2.3.25

Write each expression with a common denominator of $324$ , by multiplying each by an appropriate factor of $1$ .

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Step 10.2.3.25.1

Multiply $\frac{5}{36}$ by $\frac{9}{9}$ .

$\frac{5 \cdot 9}{36 \cdot 9} - \frac{19}{324}$

Step 10.2.3.25.2

Multiply $36$ by $9$ .

$\frac{5 \cdot 9}{324} - \frac{19}{324}$

Step 10.2.3.26

Combine the numerators over the common denominator.

$\frac{5 \cdot 9 - 19}{324}$

Step 10.2.3.27

Simplify the numerator.

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Step 10.2.3.27.1

Multiply $5$ by $9$ .

$\frac{45 - 19}{324}$

Step 10.2.3.27.2

Subtract $19$ from $45$ .

$\frac{26}{324}$

Step 10.2.3.28

Cancel the common factor of $26$ and $324$ .

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Step 10.2.3.28.1

Factor $2$ out of $26$ .

$\frac{2 (13)}{324}$

Step 10.2.3.28.2

Cancel the common factors.

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Step 10.2.3.28.2.1

Factor $2$ out of $324$ .

$\frac{2 \cdot 13}{2 \cdot 162}$

Step 10.2.3.28.2.2

Cancel the common factor.

$\frac{2 \cdot 13}{2 \cdot 162}$

Step 10.2.3.28.2.3

Rewrite the expression.

$\frac{13}{162}$

Step 11

The result can be shown in multiple forms.

Exact Form:

$\frac{13}{162}$

Decimal Form:

$0.08024691 \dots$

Step 12