Calculus Examples

$\int x^{2} {(\frac{x^{3}}{18} - 1)}^{5} d x$

Step 1

Simplify.

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Step 1.1

Use the Binomial Theorem.

$\int x^{2} ({(\frac{x^{3}}{18})}^{5} + 5 {(\frac{x^{3}}{18})}^{4} \cdot - 1 + 10 {(\frac{x^{3}}{18})}^{3} {(- 1)}^{2} + 10 {(\frac{x^{3}}{18})}^{2} {(- 1)}^{3} + 5 \frac{x^{3}}{18} {(- 1)}^{4} + {(- 1)}^{5}) d x$

Step 1.2

Simplify each term.

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Step 1.2.1

Apply the product rule to $\frac{x^{3}}{18}$ .

$\int x^{2} ({(\frac{x^{3}}{18})}^{5} + 5 \frac{{(x^{3})}^{4}}{18^{4}} \cdot - 1 + 10 {(\frac{x^{3}}{18})}^{3} {(- 1)}^{2} + 10 {(\frac{x^{3}}{18})}^{2} {(- 1)}^{3} + 5 \frac{x^{3}}{18} {(- 1)}^{4} + {(- 1)}^{5}) d x$

Step 1.2.2

Multiply the exponents in ${(x^{3})}^{4}$ .

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Step 1.2.2.1

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

$\int x^{2} ({(\frac{x^{3}}{18})}^{5} + 5 \frac{x^{3 \cdot 4}}{18^{4}} \cdot - 1 + 10 {(\frac{x^{3}}{18})}^{3} {(- 1)}^{2} + 10 {(\frac{x^{3}}{18})}^{2} {(- 1)}^{3} + 5 \frac{x^{3}}{18} {(- 1)}^{4} + {(- 1)}^{5}) d x$

Step 1.2.2.2

Multiply $3$ by $4$ .

$\int x^{2} ({(\frac{x^{3}}{18})}^{5} + 5 \frac{x^{12}}{18^{4}} \cdot - 1 + 10 {(\frac{x^{3}}{18})}^{3} {(- 1)}^{2} + 10 {(\frac{x^{3}}{18})}^{2} {(- 1)}^{3} + 5 \frac{x^{3}}{18} {(- 1)}^{4} + {(- 1)}^{5}) d x$

Step 1.2.3

Raise $18$ to the power of $4$ .

$\int x^{2} ({(\frac{x^{3}}{18})}^{5} + 5 \frac{x^{12}}{104976} \cdot - 1 + 10 {(\frac{x^{3}}{18})}^{3} {(- 1)}^{2} + 10 {(\frac{x^{3}}{18})}^{2} {(- 1)}^{3} + 5 \frac{x^{3}}{18} {(- 1)}^{4} + {(- 1)}^{5}) d x$

Step 1.2.4

Combine $5$ and $\frac{x^{12}}{104976}$ .

$\int x^{2} ({(\frac{x^{3}}{18})}^{5} + \frac{5 x^{12}}{104976} \cdot - 1 + 10 {(\frac{x^{3}}{18})}^{3} {(- 1)}^{2} + 10 {(\frac{x^{3}}{18})}^{2} {(- 1)}^{3} + 5 \frac{x^{3}}{18} {(- 1)}^{4} + {(- 1)}^{5}) d x$

Step 1.2.5

Multiply $\frac{5 x^{12}}{104976} \cdot - 1$ .

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Step 1.2.5.1

Combine $\frac{5 x^{12}}{104976}$ and $- 1$ .

$\int x^{2} ({(\frac{x^{3}}{18})}^{5} + \frac{5 x^{12} \cdot - 1}{104976} + 10 {(\frac{x^{3}}{18})}^{3} {(- 1)}^{2} + 10 {(\frac{x^{3}}{18})}^{2} {(- 1)}^{3} + 5 \frac{x^{3}}{18} {(- 1)}^{4} + {(- 1)}^{5}) d x$

Step 1.2.5.2

Multiply $- 1$ by $5$ .

$\int x^{2} ({(\frac{x^{3}}{18})}^{5} + \frac{- 5 x^{12}}{104976} + 10 {(\frac{x^{3}}{18})}^{3} {(- 1)}^{2} + 10 {(\frac{x^{3}}{18})}^{2} {(- 1)}^{3} + 5 \frac{x^{3}}{18} {(- 1)}^{4} + {(- 1)}^{5}) d x$

Step 1.2.6

Move the negative in front of the fraction.

$\int x^{2} ({(\frac{x^{3}}{18})}^{5} - \frac{5 x^{12}}{104976} + 10 {(\frac{x^{3}}{18})}^{3} {(- 1)}^{2} + 10 {(\frac{x^{3}}{18})}^{2} {(- 1)}^{3} + 5 \frac{x^{3}}{18} {(- 1)}^{4} + {(- 1)}^{5}) d x$

Step 1.2.7

Apply the product rule to $\frac{x^{3}}{18}$ .

$\int x^{2} ({(\frac{x^{3}}{18})}^{5} - \frac{5 x^{12}}{104976} + 10 \frac{{(x^{3})}^{3}}{18^{3}} {(- 1)}^{2} + 10 {(\frac{x^{3}}{18})}^{2} {(- 1)}^{3} + 5 \frac{x^{3}}{18} {(- 1)}^{4} + {(- 1)}^{5}) d x$

Step 1.2.8

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

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Step 1.2.8.1

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

$\int x^{2} ({(\frac{x^{3}}{18})}^{5} - \frac{5 x^{12}}{104976} + 10 \frac{x^{3 \cdot 3}}{18^{3}} {(- 1)}^{2} + 10 {(\frac{x^{3}}{18})}^{2} {(- 1)}^{3} + 5 \frac{x^{3}}{18} {(- 1)}^{4} + {(- 1)}^{5}) d x$

Step 1.2.8.2

Multiply $3$ by $3$ .

$\int x^{2} ({(\frac{x^{3}}{18})}^{5} - \frac{5 x^{12}}{104976} + 10 \frac{x^{9}}{18^{3}} {(- 1)}^{2} + 10 {(\frac{x^{3}}{18})}^{2} {(- 1)}^{3} + 5 \frac{x^{3}}{18} {(- 1)}^{4} + {(- 1)}^{5}) d x$

Step 1.2.9

Raise $18$ to the power of $3$ .

$\int x^{2} ({(\frac{x^{3}}{18})}^{5} - \frac{5 x^{12}}{104976} + 10 \frac{x^{9}}{5832} {(- 1)}^{2} + 10 {(\frac{x^{3}}{18})}^{2} {(- 1)}^{3} + 5 \frac{x^{3}}{18} {(- 1)}^{4} + {(- 1)}^{5}) d x$

Step 1.2.10

Cancel the common factor of $2$ .

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Step 1.2.10.1

Factor $2$ out of $10$ .

$\int x^{2} ({(\frac{x^{3}}{18})}^{5} - \frac{5 x^{12}}{104976} + 2 (5) \frac{x^{9}}{5832} {(- 1)}^{2} + 10 {(\frac{x^{3}}{18})}^{2} {(- 1)}^{3} + 5 \frac{x^{3}}{18} {(- 1)}^{4} + {(- 1)}^{5}) d x$

Step 1.2.10.2

Factor $2$ out of $5832$ .

$\int x^{2} ({(\frac{x^{3}}{18})}^{5} - \frac{5 x^{12}}{104976} + 2 \cdot 5 \frac{x^{9}}{2 \cdot 2916} {(- 1)}^{2} + 10 {(\frac{x^{3}}{18})}^{2} {(- 1)}^{3} + 5 \frac{x^{3}}{18} {(- 1)}^{4} + {(- 1)}^{5}) d x$

Step 1.2.10.3

Cancel the common factor.

Step 1.2.10.4

Rewrite the expression.

$\int x^{2} ({(\frac{x^{3}}{18})}^{5} - \frac{5 x^{12}}{104976} + 5 \frac{x^{9}}{2916} {(- 1)}^{2} + 10 {(\frac{x^{3}}{18})}^{2} {(- 1)}^{3} + 5 \frac{x^{3}}{18} {(- 1)}^{4} + {(- 1)}^{5}) d x$

Step 1.2.11

Combine $5$ and $\frac{x^{9}}{2916}$ .

$\int x^{2} ({(\frac{x^{3}}{18})}^{5} - \frac{5 x^{12}}{104976} + \frac{5 x^{9}}{2916} {(- 1)}^{2} + 10 {(\frac{x^{3}}{18})}^{2} {(- 1)}^{3} + 5 \frac{x^{3}}{18} {(- 1)}^{4} + {(- 1)}^{5}) d x$

Step 1.2.12

Raise $- 1$ to the power of $2$ .

$\int x^{2} ({(\frac{x^{3}}{18})}^{5} - \frac{5 x^{12}}{104976} + \frac{5 x^{9}}{2916} \cdot 1 + 10 {(\frac{x^{3}}{18})}^{2} {(- 1)}^{3} + 5 \frac{x^{3}}{18} {(- 1)}^{4} + {(- 1)}^{5}) d x$

Step 1.2.13

Multiply $\frac{5 x^{9}}{2916}$ by $1$ .

$\int x^{2} ({(\frac{x^{3}}{18})}^{5} - \frac{5 x^{12}}{104976} + \frac{5 x^{9}}{2916} + 10 {(\frac{x^{3}}{18})}^{2} {(- 1)}^{3} + 5 \frac{x^{3}}{18} {(- 1)}^{4} + {(- 1)}^{5}) d x$

Step 1.2.14

Apply the product rule to $\frac{x^{3}}{18}$ .

$\int x^{2} ({(\frac{x^{3}}{18})}^{5} - \frac{5 x^{12}}{104976} + \frac{5 x^{9}}{2916} + 10 \frac{{(x^{3})}^{2}}{18^{2}} {(- 1)}^{3} + 5 \frac{x^{3}}{18} {(- 1)}^{4} + {(- 1)}^{5}) d x$

Step 1.2.15

Multiply the exponents in ${(x^{3})}^{2}$ .

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Step 1.2.15.1

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

$\int x^{2} ({(\frac{x^{3}}{18})}^{5} - \frac{5 x^{12}}{104976} + \frac{5 x^{9}}{2916} + 10 \frac{x^{3 \cdot 2}}{18^{2}} {(- 1)}^{3} + 5 \frac{x^{3}}{18} {(- 1)}^{4} + {(- 1)}^{5}) d x$

Step 1.2.15.2

Multiply $3$ by $2$ .

$\int x^{2} ({(\frac{x^{3}}{18})}^{5} - \frac{5 x^{12}}{104976} + \frac{5 x^{9}}{2916} + 10 \frac{x^{6}}{18^{2}} {(- 1)}^{3} + 5 \frac{x^{3}}{18} {(- 1)}^{4} + {(- 1)}^{5}) d x$

Step 1.2.16

Raise $18$ to the power of $2$ .

$\int x^{2} ({(\frac{x^{3}}{18})}^{5} - \frac{5 x^{12}}{104976} + \frac{5 x^{9}}{2916} + 10 \frac{x^{6}}{324} {(- 1)}^{3} + 5 \frac{x^{3}}{18} {(- 1)}^{4} + {(- 1)}^{5}) d x$

Step 1.2.17

Cancel the common factor of $2$ .

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Step 1.2.17.1

Factor $2$ out of $10$ .

$\int x^{2} ({(\frac{x^{3}}{18})}^{5} - \frac{5 x^{12}}{104976} + \frac{5 x^{9}}{2916} + 2 (5) \frac{x^{6}}{324} {(- 1)}^{3} + 5 \frac{x^{3}}{18} {(- 1)}^{4} + {(- 1)}^{5}) d x$

Step 1.2.17.2

Factor $2$ out of $324$ .

$\int x^{2} ({(\frac{x^{3}}{18})}^{5} - \frac{5 x^{12}}{104976} + \frac{5 x^{9}}{2916} + 2 \cdot 5 \frac{x^{6}}{2 \cdot 162} {(- 1)}^{3} + 5 \frac{x^{3}}{18} {(- 1)}^{4} + {(- 1)}^{5}) d x$

Step 1.2.17.3

Cancel the common factor.

$\int x^{2} ({(\frac{x^{3}}{18})}^{5} - \frac{5 x^{12}}{104976} + \frac{5 x^{9}}{2916} + 2 \cdot 5 \frac{x^{6}}{2 \cdot 162} {(- 1)}^{3} + 5 \frac{x^{3}}{18} {(- 1)}^{4} + {(- 1)}^{5}) d x$

Step 1.2.17.4

Rewrite the expression.

$\int x^{2} ({(\frac{x^{3}}{18})}^{5} - \frac{5 x^{12}}{104976} + \frac{5 x^{9}}{2916} + 5 \frac{x^{6}}{162} {(- 1)}^{3} + 5 \frac{x^{3}}{18} {(- 1)}^{4} + {(- 1)}^{5}) d x$

Step 1.2.18

Combine $5$ and $\frac{x^{6}}{162}$ .

$\int x^{2} ({(\frac{x^{3}}{18})}^{5} - \frac{5 x^{12}}{104976} + \frac{5 x^{9}}{2916} + \frac{5 x^{6}}{162} {(- 1)}^{3} + 5 \frac{x^{3}}{18} {(- 1)}^{4} + {(- 1)}^{5}) d x$

Step 1.2.19

Raise $- 1$ to the power of $3$ .

$\int x^{2} ({(\frac{x^{3}}{18})}^{5} - \frac{5 x^{12}}{104976} + \frac{5 x^{9}}{2916} + \frac{5 x^{6}}{162} \cdot - 1 + 5 \frac{x^{3}}{18} {(- 1)}^{4} + {(- 1)}^{5}) d x$

Step 1.2.20

Multiply $\frac{5 x^{6}}{162} \cdot - 1$ .

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Step 1.2.20.1

Combine $\frac{5 x^{6}}{162}$ and $- 1$ .

$\int x^{2} ({(\frac{x^{3}}{18})}^{5} - \frac{5 x^{12}}{104976} + \frac{5 x^{9}}{2916} + \frac{5 x^{6} \cdot - 1}{162} + 5 \frac{x^{3}}{18} {(- 1)}^{4} + {(- 1)}^{5}) d x$

Step 1.2.20.2

Multiply $- 1$ by $5$ .

$\int x^{2} ({(\frac{x^{3}}{18})}^{5} - \frac{5 x^{12}}{104976} + \frac{5 x^{9}}{2916} + \frac{- 5 x^{6}}{162} + 5 \frac{x^{3}}{18} {(- 1)}^{4} + {(- 1)}^{5}) d x$

Step 1.2.21

Move the negative in front of the fraction.

$\int x^{2} ({(\frac{x^{3}}{18})}^{5} - \frac{5 x^{12}}{104976} + \frac{5 x^{9}}{2916} - \frac{5 x^{6}}{162} + 5 \frac{x^{3}}{18} {(- 1)}^{4} + {(- 1)}^{5}) d x$

Step 1.2.22

Combine $5$ and $\frac{x^{3}}{18}$ .

$\int x^{2} ({(\frac{x^{3}}{18})}^{5} - \frac{5 x^{12}}{104976} + \frac{5 x^{9}}{2916} - \frac{5 x^{6}}{162} + \frac{5 x^{3}}{18} {(- 1)}^{4} + {(- 1)}^{5}) d x$

Step 1.2.23

Raise $- 1$ to the power of $4$ .

$\int x^{2} ({(\frac{x^{3}}{18})}^{5} - \frac{5 x^{12}}{104976} + \frac{5 x^{9}}{2916} - \frac{5 x^{6}}{162} + \frac{5 x^{3}}{18} \cdot 1 + {(- 1)}^{5}) d x$

Step 1.2.24

Multiply $\frac{5 x^{3}}{18}$ by $1$ .

$\int x^{2} ({(\frac{x^{3}}{18})}^{5} - \frac{5 x^{12}}{104976} + \frac{5 x^{9}}{2916} - \frac{5 x^{6}}{162} + \frac{5 x^{3}}{18} + {(- 1)}^{5}) d x$

Step 1.2.25

Raise $- 1$ to the power of $5$ .

$\int x^{2} ({(\frac{x^{3}}{18})}^{5} - \frac{5 x^{12}}{104976} + \frac{5 x^{9}}{2916} - \frac{5 x^{6}}{162} + \frac{5 x^{3}}{18} - 1) d x$

Step 1.3

Apply the distributive property.

$\int x^{2} {(\frac{x^{3}}{18})}^{5} + x^{2} (- \frac{5 x^{12}}{104976}) + x^{2} \frac{5 x^{9}}{2916} + x^{2} (- \frac{5 x^{6}}{162}) + x^{2} \frac{5 x^{3}}{18} + x^{2} \cdot - 1 d x$

Step 1.4

Simplify.

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Step 1.4.1

Rewrite using the commutative property of multiplication.

$\int x^{2} {(\frac{x^{3}}{18})}^{5} - x^{2} \frac{5 x^{12}}{104976} + x^{2} \frac{5 x^{9}}{2916} + x^{2} (- \frac{5 x^{6}}{162}) + x^{2} \frac{5 x^{3}}{18} + x^{2} \cdot - 1 d x$

Step 1.4.2

Multiply $x^{2} \frac{5 x^{9}}{2916}$ .

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Step 1.4.2.1

Combine $x^{2}$ and $\frac{5 x^{9}}{2916}$ .

$\int x^{2} {(\frac{x^{3}}{18})}^{5} - x^{2} \frac{5 x^{12}}{104976} + \frac{x^{2} (5 x^{9})}{2916} + x^{2} (- \frac{5 x^{6}}{162}) + x^{2} \frac{5 x^{3}}{18} + x^{2} \cdot - 1 d x$

Step 1.4.2.2

Multiply $x^{2}$ by $x^{9}$ by adding the exponents.

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Step 1.4.2.2.1

Move $x^{9}$ .

$\int x^{2} {(\frac{x^{3}}{18})}^{5} - x^{2} \frac{5 x^{12}}{104976} + \frac{x^{9} x^{2} \cdot 5}{2916} + x^{2} (- \frac{5 x^{6}}{162}) + x^{2} \frac{5 x^{3}}{18} + x^{2} \cdot - 1 d x$

Step 1.4.2.2.2

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

$\int x^{2} {(\frac{x^{3}}{18})}^{5} - x^{2} \frac{5 x^{12}}{104976} + \frac{x^{9 + 2} \cdot 5}{2916} + x^{2} (- \frac{5 x^{6}}{162}) + x^{2} \frac{5 x^{3}}{18} + x^{2} \cdot - 1 d x$

Step 1.4.2.2.3

Add $9$ and $2$ .

$\int x^{2} {(\frac{x^{3}}{18})}^{5} - x^{2} \frac{5 x^{12}}{104976} + \frac{x^{11} \cdot 5}{2916} + x^{2} (- \frac{5 x^{6}}{162}) + x^{2} \frac{5 x^{3}}{18} + x^{2} \cdot - 1 d x$

Step 1.4.3

Rewrite using the commutative property of multiplication.

$\int x^{2} {(\frac{x^{3}}{18})}^{5} - x^{2} \frac{5 x^{12}}{104976} + \frac{x^{11} \cdot 5}{2916} - x^{2} \frac{5 x^{6}}{162} + x^{2} \frac{5 x^{3}}{18} + x^{2} \cdot - 1 d x$

Step 1.4.4

Multiply $x^{2} \frac{5 x^{3}}{18}$ .

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Step 1.4.4.1

Combine $x^{2}$ and $\frac{5 x^{3}}{18}$ .

$\int x^{2} {(\frac{x^{3}}{18})}^{5} - x^{2} \frac{5 x^{12}}{104976} + \frac{x^{11} \cdot 5}{2916} - x^{2} \frac{5 x^{6}}{162} + \frac{x^{2} (5 x^{3})}{18} + x^{2} \cdot - 1 d x$

Step 1.4.4.2

Multiply $x^{2}$ by $x^{3}$ by adding the exponents.

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Step 1.4.4.2.1

Move $x^{3}$ .

$\int x^{2} {(\frac{x^{3}}{18})}^{5} - x^{2} \frac{5 x^{12}}{104976} + \frac{x^{11} \cdot 5}{2916} - x^{2} \frac{5 x^{6}}{162} + \frac{x^{3} x^{2} \cdot 5}{18} + x^{2} \cdot - 1 d x$

Step 1.4.4.2.2

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

$\int x^{2} {(\frac{x^{3}}{18})}^{5} - x^{2} \frac{5 x^{12}}{104976} + \frac{x^{11} \cdot 5}{2916} - x^{2} \frac{5 x^{6}}{162} + \frac{x^{3 + 2} \cdot 5}{18} + x^{2} \cdot - 1 d x$

Step 1.4.4.2.3

Add $3$ and $2$ .

$\int x^{2} {(\frac{x^{3}}{18})}^{5} - x^{2} \frac{5 x^{12}}{104976} + \frac{x^{11} \cdot 5}{2916} - x^{2} \frac{5 x^{6}}{162} + \frac{x^{5} \cdot 5}{18} + x^{2} \cdot - 1 d x$

Step 1.4.5

Move $- 1$ to the left of $x^{2}$ .

$\int x^{2} {(\frac{x^{3}}{18})}^{5} - x^{2} \frac{5 x^{12}}{104976} + \frac{x^{11} \cdot 5}{2916} - x^{2} \frac{5 x^{6}}{162} + \frac{x^{5} \cdot 5}{18} - 1 \cdot x^{2} d x$

Step 1.5

Simplify each term.

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Step 1.5.1

Apply the product rule to $\frac{x^{3}}{18}$ .

$\int x^{2} \frac{{(x^{3})}^{5}}{18^{5}} - x^{2} \frac{5 x^{12}}{104976} + \frac{x^{11} \cdot 5}{2916} - x^{2} \frac{5 x^{6}}{162} + \frac{x^{5} \cdot 5}{18} - 1 \cdot x^{2} d x$

Step 1.5.2

Multiply the exponents in ${(x^{3})}^{5}$ .

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Step 1.5.2.1

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

$\int x^{2} \frac{x^{3 \cdot 5}}{18^{5}} - x^{2} \frac{5 x^{12}}{104976} + \frac{x^{11} \cdot 5}{2916} - x^{2} \frac{5 x^{6}}{162} + \frac{x^{5} \cdot 5}{18} - 1 \cdot x^{2} d x$

Step 1.5.2.2

Multiply $3$ by $5$ .

$\int x^{2} \frac{x^{15}}{18^{5}} - x^{2} \frac{5 x^{12}}{104976} + \frac{x^{11} \cdot 5}{2916} - x^{2} \frac{5 x^{6}}{162} + \frac{x^{5} \cdot 5}{18} - 1 \cdot x^{2} d x$

Step 1.5.3

Raise $18$ to the power of $5$ .

$\int x^{2} \frac{x^{15}}{1889568} - x^{2} \frac{5 x^{12}}{104976} + \frac{x^{11} \cdot 5}{2916} - x^{2} \frac{5 x^{6}}{162} + \frac{x^{5} \cdot 5}{18} - 1 \cdot x^{2} d x$

Step 1.5.4

Multiply $x^{2} \frac{x^{15}}{1889568}$ .

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Step 1.5.4.1

Combine $x^{2}$ and $\frac{x^{15}}{1889568}$ .

$\int \frac{x^{2} x^{15}}{1889568} - x^{2} \frac{5 x^{12}}{104976} + \frac{x^{11} \cdot 5}{2916} - x^{2} \frac{5 x^{6}}{162} + \frac{x^{5} \cdot 5}{18} - 1 \cdot x^{2} d x$

Step 1.5.4.2

Multiply $x^{2}$ by $x^{15}$ by adding the exponents.

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Step 1.5.4.2.1

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

$\int \frac{x^{2 + 15}}{1889568} - x^{2} \frac{5 x^{12}}{104976} + \frac{x^{11} \cdot 5}{2916} - x^{2} \frac{5 x^{6}}{162} + \frac{x^{5} \cdot 5}{18} - 1 \cdot x^{2} d x$

Step 1.5.4.2.2

Add $2$ and $15$ .

$\int \frac{x^{17}}{1889568} - x^{2} \frac{5 x^{12}}{104976} + \frac{x^{11} \cdot 5}{2916} - x^{2} \frac{5 x^{6}}{162} + \frac{x^{5} \cdot 5}{18} - 1 \cdot x^{2} d x$

Step 1.5.5

Multiply $- x^{2} \frac{5 x^{12}}{104976}$ .

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Step 1.5.5.1

Combine $\frac{5 x^{12}}{104976}$ and $x^{2}$ .

$\int \frac{x^{17}}{1889568} - \frac{5 x^{12} x^{2}}{104976} + \frac{x^{11} \cdot 5}{2916} - x^{2} \frac{5 x^{6}}{162} + \frac{x^{5} \cdot 5}{18} - 1 \cdot x^{2} d x$

Step 1.5.5.2

Multiply $x^{12}$ by $x^{2}$ by adding the exponents.

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Step 1.5.5.2.1

Move $x^{2}$ .

$\int \frac{x^{17}}{1889568} - \frac{5 (x^{2} x^{12})}{104976} + \frac{x^{11} \cdot 5}{2916} - x^{2} \frac{5 x^{6}}{162} + \frac{x^{5} \cdot 5}{18} - 1 \cdot x^{2} d x$

Step 1.5.5.2.2

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

$\int \frac{x^{17}}{1889568} - \frac{5 x^{2 + 12}}{104976} + \frac{x^{11} \cdot 5}{2916} - x^{2} \frac{5 x^{6}}{162} + \frac{x^{5} \cdot 5}{18} - 1 \cdot x^{2} d x$

Step 1.5.5.2.3

Add $2$ and $12$ .

$\int \frac{x^{17}}{1889568} - \frac{5 x^{14}}{104976} + \frac{x^{11} \cdot 5}{2916} - x^{2} \frac{5 x^{6}}{162} + \frac{x^{5} \cdot 5}{18} - 1 \cdot x^{2} d x$

Step 1.5.6

Move $5$ to the left of $x^{11}$ .

$\int \frac{x^{17}}{1889568} - \frac{5 x^{14}}{104976} + \frac{5 \cdot x^{11}}{2916} - x^{2} \frac{5 x^{6}}{162} + \frac{x^{5} \cdot 5}{18} - 1 \cdot x^{2} d x$

Step 1.5.7

Multiply $- x^{2} \frac{5 x^{6}}{162}$ .

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Step 1.5.7.1

Combine $\frac{5 x^{6}}{162}$ and $x^{2}$ .

$\int \frac{x^{17}}{1889568} - \frac{5 x^{14}}{104976} + \frac{5 x^{11}}{2916} - \frac{5 x^{6} x^{2}}{162} + \frac{x^{5} \cdot 5}{18} - 1 \cdot x^{2} d x$

Step 1.5.7.2

Multiply $x^{6}$ by $x^{2}$ by adding the exponents.

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Step 1.5.7.2.1

Move $x^{2}$ .

$\int \frac{x^{17}}{1889568} - \frac{5 x^{14}}{104976} + \frac{5 x^{11}}{2916} - \frac{5 (x^{2} x^{6})}{162} + \frac{x^{5} \cdot 5}{18} - 1 \cdot x^{2} d x$

Step 1.5.7.2.2

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

$\int \frac{x^{17}}{1889568} - \frac{5 x^{14}}{104976} + \frac{5 x^{11}}{2916} - \frac{5 x^{2 + 6}}{162} + \frac{x^{5} \cdot 5}{18} - 1 \cdot x^{2} d x$

Step 1.5.7.2.3

Add $2$ and $6$ .

$\int \frac{x^{17}}{1889568} - \frac{5 x^{14}}{104976} + \frac{5 x^{11}}{2916} - \frac{5 x^{8}}{162} + \frac{x^{5} \cdot 5}{18} - 1 \cdot x^{2} d x$

Step 1.5.8

Move $5$ to the left of $x^{5}$ .

$\int \frac{x^{17}}{1889568} - \frac{5 x^{14}}{104976} + \frac{5 x^{11}}{2916} - \frac{5 x^{8}}{162} + \frac{5 \cdot x^{5}}{18} - 1 \cdot x^{2} d x$

Step 1.5.9

Rewrite $- 1 x^{2}$ as $- x^{2}$ .

$\int \frac{x^{17}}{1889568} - \frac{5 x^{14}}{104976} + \frac{5 x^{11}}{2916} - \frac{5 x^{8}}{162} + \frac{5 x^{5}}{18} - x^{2} d x$

Step 2

Split the single integral into multiple integrals.

$\int \frac{x^{17}}{1889568} d x + \int - \frac{5 x^{14}}{104976} d x + \int \frac{5 x^{11}}{2916} d x + \int - \frac{5 x^{8}}{162} d x + \int \frac{5 x^{5}}{18} d x + \int - x^{2} d x$

Step 3

Since $\frac{1}{1889568}$ is constant with respect to $x$ , move $\frac{1}{1889568}$ out of the integral.

$\frac{1}{1889568} \int x^{17} d x + \int - \frac{5 x^{14}}{104976} d x + \int \frac{5 x^{11}}{2916} d x + \int - \frac{5 x^{8}}{162} d x + \int \frac{5 x^{5}}{18} d x + \int - x^{2} d x$

Step 4

By the Power Rule, the integral of $x^{17}$ with respect to $x$ is $\frac{1}{18} x^{18}$ .

$\frac{1}{1889568} (\frac{1}{18} x^{18} + C) + \int - \frac{5 x^{14}}{104976} d x + \int \frac{5 x^{11}}{2916} d x + \int - \frac{5 x^{8}}{162} d x + \int \frac{5 x^{5}}{18} d x + \int - x^{2} d x$

Step 5

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

$\frac{1}{1889568} (\frac{1}{18} x^{18} + C) - \int \frac{5 x^{14}}{104976} d x + \int \frac{5 x^{11}}{2916} d x + \int - \frac{5 x^{8}}{162} d x + \int \frac{5 x^{5}}{18} d x + \int - x^{2} d x$

Step 6

Since $\frac{5}{104976}$ is constant with respect to $x$ , move $\frac{5}{104976}$ out of the integral.

$\frac{1}{1889568} (\frac{1}{18} x^{18} + C) - (\frac{5}{104976} \int x^{14} d x) + \int \frac{5 x^{11}}{2916} d x + \int - \frac{5 x^{8}}{162} d x + \int \frac{5 x^{5}}{18} d x + \int - x^{2} d x$

Step 7

By the Power Rule, the integral of $x^{14}$ with respect to $x$ is $\frac{1}{15} x^{15}$ .

$\frac{1}{1889568} (\frac{1}{18} x^{18} + C) - \frac{5}{104976} (\frac{1}{15} x^{15} + C) + \int \frac{5 x^{11}}{2916} d x + \int - \frac{5 x^{8}}{162} d x + \int \frac{5 x^{5}}{18} d x + \int - x^{2} d x$

Step 8

Since $\frac{5}{2916}$ is constant with respect to $x$ , move $\frac{5}{2916}$ out of the integral.

$\frac{1}{1889568} (\frac{1}{18} x^{18} + C) - \frac{5}{104976} (\frac{1}{15} x^{15} + C) + \frac{5}{2916} \int x^{11} d x + \int - \frac{5 x^{8}}{162} d x + \int \frac{5 x^{5}}{18} d x + \int - x^{2} d x$

Step 9

By the Power Rule, the integral of $x^{11}$ with respect to $x$ is $\frac{1}{12} x^{12}$ .

$\frac{1}{1889568} (\frac{1}{18} x^{18} + C) - \frac{5}{104976} (\frac{1}{15} x^{15} + C) + \frac{5}{2916} (\frac{1}{12} x^{12} + C) + \int - \frac{5 x^{8}}{162} d x + \int \frac{5 x^{5}}{18} d x + \int - x^{2} d x$

Step 10

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

$\frac{1}{1889568} (\frac{1}{18} x^{18} + C) - \frac{5}{104976} (\frac{1}{15} x^{15} + C) + \frac{5}{2916} (\frac{1}{12} x^{12} + C) - \int \frac{5 x^{8}}{162} d x + \int \frac{5 x^{5}}{18} d x + \int - x^{2} d x$

Step 11

Since $\frac{5}{162}$ is constant with respect to $x$ , move $\frac{5}{162}$ out of the integral.

$\frac{1}{1889568} (\frac{1}{18} x^{18} + C) - \frac{5}{104976} (\frac{1}{15} x^{15} + C) + \frac{5}{2916} (\frac{1}{12} x^{12} + C) - (\frac{5}{162} \int x^{8} d x) + \int \frac{5 x^{5}}{18} d x + \int - x^{2} d x$

Step 12

By the Power Rule, the integral of $x^{8}$ with respect to $x$ is $\frac{1}{9} x^{9}$ .

$\frac{1}{1889568} (\frac{1}{18} x^{18} + C) - \frac{5}{104976} (\frac{1}{15} x^{15} + C) + \frac{5}{2916} (\frac{1}{12} x^{12} + C) - \frac{5}{162} (\frac{1}{9} x^{9} + C) + \int \frac{5 x^{5}}{18} d x + \int - x^{2} d x$

Step 13

Since $\frac{5}{18}$ is constant with respect to $x$ , move $\frac{5}{18}$ out of the integral.

Step 14

By the Power Rule, the integral of $x^{5}$ with respect to $x$ is $\frac{1}{6} x^{6}$ .

Step 15

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

Step 16

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

Step 17

Simplify.

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Step 17.1

Simplify.

$\frac{x^{18}}{34012224} - \frac{x^{15}}{314928} + \frac{5 x^{12}}{34992} - \frac{5 x^{9}}{1458} + \frac{5 x^{6}}{108} - (\frac{1}{3} x^{3}) + C$

Step 17.2

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

$\frac{x^{18}}{34012224} - \frac{x^{15}}{314928} + \frac{5 x^{12}}{34992} - \frac{5 x^{9}}{1458} + \frac{5 x^{6}}{108} - \frac{x^{3}}{3} + C$

Step 18

Reorder terms.

$\frac{1}{34012224} x^{18} - \frac{1}{314928} x^{15} + \frac{5}{34992} x^{12} - \frac{5}{1458} x^{9} + \frac{5}{108} x^{6} - \frac{1}{3} x^{3} + C$