Calculus Examples

$\int 64 x^{3} {(4 x^{2} - 1)}^{3} d x$

Step 1

Since $64$ is constant with respect to $x$ , move $64$ out of the integral.

$64 \int x^{3} {(4 x^{2} - 1)}^{3} d x$

Step 2

Expand $x^{3} {(4 x^{2} - 1)}^{3}$ .

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Step 2.1

Use the Binomial Theorem.

$64 \int x^{3} ({(4 x^{2})}^{3} + 3 {(4 x^{2})}^{2} \cdot - 1 + 3 (4 x^{2}) {(- 1)}^{2} + {(- 1)}^{3}) d x$

Step 2.2

Rewrite the exponentiation as a product.

$64 \int x^{3} (4 x^{2} {(4 x^{2})}^{2} + 3 {(4 x^{2})}^{2} \cdot - 1 + 3 (4 x^{2}) {(- 1)}^{2} + {(- 1)}^{3}) d x$

Step 2.3

Rewrite the exponentiation as a product.

$64 \int x^{3} (4 x^{2} (4 x^{2} (4 x^{2})) + 3 {(4 x^{2})}^{2} \cdot - 1 + 3 (4 x^{2}) {(- 1)}^{2} + {(- 1)}^{3}) d x$

Step 2.4

Rewrite the exponentiation as a product.

$64 \int x^{3} (4 x^{2} (4 x^{2} (4 x^{2})) + 3 (4 x^{2} (4 x^{2})) \cdot - 1 + 3 (4 x^{2}) {(- 1)}^{2} + {(- 1)}^{3}) d x$

Step 2.5

Rewrite the exponentiation as a product.

$64 \int x^{3} (4 x^{2} (4 x^{2} (4 x^{2})) + 3 (4 x^{2} (4 x^{2})) \cdot - 1 + 3 (4 x^{2}) (- - 1) + {(- 1)}^{3}) d x$

Step 2.6

Rewrite the exponentiation as a product.

$64 \int x^{3} (4 x^{2} (4 x^{2} (4 x^{2})) + 3 (4 x^{2} (4 x^{2})) \cdot - 1 + 3 (4 x^{2}) (- - 1) - {(- 1)}^{2}) d x$

Step 2.7

Rewrite the exponentiation as a product.

$64 \int x^{3} (4 x^{2} (4 x^{2} (4 x^{2})) + 3 (4 x^{2} (4 x^{2})) \cdot - 1 + 3 (4 x^{2}) (- - 1) - - - 1) d x$

Step 2.8

Apply the distributive property.

$64 \int x^{3} (4 x^{2} (4 x^{2} (4 x^{2})) + 3 (4 x^{2} (4 x^{2})) \cdot - 1 + 3 (4 x^{2}) (- - 1)) + x^{3} (- - - 1) d x$

Step 2.9

Apply the distributive property.

$64 \int x^{3} (4 x^{2} (4 x^{2} (4 x^{2})) + 3 (4 x^{2} (4 x^{2})) \cdot - 1) + x^{3} (3 (4 x^{2}) (- - 1)) + x^{3} (- - - 1) d x$

Step 2.10

Apply the distributive property.

$64 \int x^{3} (4 x^{2} (4 x^{2} (4 x^{2}))) + x^{3} (3 (4 x^{2} (4 x^{2})) \cdot - 1) + x^{3} (3 (4 x^{2}) (- - 1)) + x^{3} (- - - 1) d x$

Step 2.11

Move $x^{2}$ .

$64 \int x^{3} (4 x^{2} (4 \cdot 4 x^{2} x^{2})) + x^{3} (3 (4 x^{2} (4 x^{2})) \cdot - 1) + x^{3} (3 (4 x^{2}) (- - 1)) + x^{3} (- - - 1) d x$

Step 2.12

Move parentheses.

$64 \int x^{3} (4 x^{2} (4 \cdot 4 x^{2}) x^{2}) + x^{3} (3 (4 x^{2} (4 x^{2})) \cdot - 1) + x^{3} (3 (4 x^{2}) (- - 1)) + x^{3} (- - - 1) d x$

Step 2.13

Move parentheses.

$64 \int x^{3} (4 x^{2} (4 \cdot 4) x^{2} x^{2}) + x^{3} (3 (4 x^{2} (4 x^{2})) \cdot - 1) + x^{3} (3 (4 x^{2}) (- - 1)) + x^{3} (- - - 1) d x$

Step 2.14

Move $x^{2}$ .

$64 \int x^{3} (4 \cdot 4 \cdot 4 x^{2} x^{2} x^{2}) + x^{3} (3 (4 x^{2} (4 x^{2})) \cdot - 1) + x^{3} (3 (4 x^{2}) (- - 1)) + x^{3} (- - - 1) d x$

Step 2.15

Move parentheses.

$64 \int x^{3} (4 \cdot 4 \cdot 4 x^{2} x^{2}) x^{2} + x^{3} (3 (4 x^{2} (4 x^{2})) \cdot - 1) + x^{3} (3 (4 x^{2}) (- - 1)) + x^{3} (- - - 1) d x$

Step 2.16

Move parentheses.

$64 \int x^{3} (4 \cdot 4 \cdot 4 x^{2}) x^{2} x^{2} + x^{3} (3 (4 x^{2} (4 x^{2})) \cdot - 1) + x^{3} (3 (4 x^{2}) (- - 1)) + x^{3} (- - - 1) d x$

Step 2.17

Move parentheses.

$64 \int x^{3} (4 \cdot 4 \cdot 4) x^{2} x^{2} x^{2} + x^{3} (3 (4 x^{2} (4 x^{2})) \cdot - 1) + x^{3} (3 (4 x^{2}) (- - 1)) + x^{3} (- - - 1) d x$

Step 2.18

Move parentheses.

$64 \int x^{3} (4 \cdot 4) \cdot 4 x^{2} x^{2} x^{2} + x^{3} (3 (4 x^{2} (4 x^{2})) \cdot - 1) + x^{3} (3 (4 x^{2}) (- - 1)) + x^{3} (- - - 1) d x$

Step 2.19

Move $x^{3}$ .

$64 \int (4 \cdot 4) \cdot 4 x^{3} x^{2} x^{2} x^{2} + x^{3} (3 (4 x^{2} (4 x^{2})) \cdot - 1) + x^{3} (3 (4 x^{2}) (- - 1)) + x^{3} (- - - 1) d x$

Step 2.20

Move $x^{2}$ .

$64 \int (4 \cdot 4) \cdot 4 x^{3} x^{2} x^{2} x^{2} + x^{3} (3 (4 \cdot 4 x^{2} x^{2}) \cdot - 1) + x^{3} (3 (4 x^{2}) (- - 1)) + x^{3} (- - - 1) d x$

Step 2.21

Move parentheses.

$64 \int (4 \cdot 4) \cdot 4 x^{3} x^{2} x^{2} x^{2} + x^{3} (3 (4 \cdot 4 x^{2}) x^{2} \cdot - 1) + x^{3} (3 (4 x^{2}) (- - 1)) + x^{3} (- - - 1) d x$

Step 2.22

Move parentheses.

$64 \int (4 \cdot 4) \cdot 4 x^{3} x^{2} x^{2} x^{2} + x^{3} (3 (4 \cdot 4) x^{2} x^{2} \cdot - 1) + x^{3} (3 (4 x^{2}) (- - 1)) + x^{3} (- - - 1) d x$

Step 2.23

Move $x^{2}$ .

$64 \int (4 \cdot 4) \cdot 4 x^{3} x^{2} x^{2} x^{2} + x^{3} (3 \cdot 4 \cdot 4 x^{2} \cdot - 1 x^{2}) + x^{3} (3 (4 x^{2}) (- - 1)) + x^{3} (- - - 1) d x$

Step 2.24

Move $x^{2}$ .

$64 \int (4 \cdot 4) \cdot 4 x^{3} x^{2} x^{2} x^{2} + x^{3} (3 \cdot 4 \cdot 4 \cdot - 1 x^{2} x^{2}) + x^{3} (3 (4 x^{2}) (- - 1)) + x^{3} (- - - 1) d x$

Step 2.25

Move parentheses.

$64 \int (4 \cdot 4) \cdot 4 x^{3} x^{2} x^{2} x^{2} + x^{3} (3 \cdot 4 \cdot 4 \cdot - 1 x^{2}) x^{2} + x^{3} (3 (4 x^{2}) (- - 1)) + x^{3} (- - - 1) d x$

Step 2.26

Move parentheses.

$64 \int (4 \cdot 4) \cdot 4 x^{3} x^{2} x^{2} x^{2} + x^{3} (3 \cdot 4 \cdot 4 \cdot - 1) x^{2} x^{2} + x^{3} (3 (4 x^{2}) (- - 1)) + x^{3} (- - - 1) d x$

Step 2.27

Move parentheses.

$64 \int (4 \cdot 4) \cdot 4 x^{3} x^{2} x^{2} x^{2} + x^{3} (3 \cdot 4 \cdot 4) \cdot - 1 x^{2} x^{2} + x^{3} (3 (4 x^{2}) (- - 1)) + x^{3} (- - - 1) d x$

Step 2.28

Move $x^{3}$ .

$64 \int (4 \cdot 4) \cdot 4 x^{3} x^{2} x^{2} x^{2} + (3 \cdot 4 \cdot 4) \cdot - 1 x^{3} x^{2} x^{2} + x^{3} (3 (4 x^{2}) (- - 1)) + x^{3} (- - - 1) d x$

Step 2.29

Move $x^{2}$ .

$64 \int (4 \cdot 4) \cdot 4 x^{3} x^{2} x^{2} x^{2} + (3 \cdot 4 \cdot 4) \cdot - 1 x^{3} x^{2} x^{2} + x^{3} (3 \cdot 4 \cdot - 1 \cdot - 1 x^{2}) + x^{3} (- - - 1) d x$

Step 2.30

Move parentheses.

$64 \int (4 \cdot 4) \cdot 4 x^{3} x^{2} x^{2} x^{2} + (3 \cdot 4 \cdot 4) \cdot - 1 x^{3} x^{2} x^{2} + x^{3} (3 \cdot 4 \cdot - 1 \cdot - 1) x^{2} + x^{3} (- - - 1) d x$

Step 2.31

Move parentheses.

$64 \int (4 \cdot 4) \cdot 4 x^{3} x^{2} x^{2} x^{2} + (3 \cdot 4 \cdot 4) \cdot - 1 x^{3} x^{2} x^{2} + x^{3} (3 \cdot 4 \cdot - 1) \cdot - 1 x^{2} + x^{3} (- - - 1) d x$

Step 2.32

Move $x^{3}$ .

$64 \int (4 \cdot 4) \cdot 4 x^{3} x^{2} x^{2} x^{2} + (3 \cdot 4 \cdot 4) \cdot - 1 x^{3} x^{2} x^{2} + (3 \cdot 4 \cdot - 1) \cdot - 1 x^{3} x^{2} + x^{3} (- - - 1) d x$

Step 2.33

Move parentheses.

$64 \int (4 \cdot 4) \cdot 4 x^{3} x^{2} x^{2} x^{2} + (3 \cdot 4 \cdot 4) \cdot - 1 x^{3} x^{2} x^{2} + (3 \cdot 4 \cdot - 1) \cdot - 1 x^{3} x^{2} + x^{3} (- - 1 \cdot - 1) d x$

Step 2.34

Move parentheses.

$64 \int (4 \cdot 4) \cdot 4 x^{3} x^{2} x^{2} x^{2} + (3 \cdot 4 \cdot 4) \cdot - 1 x^{3} x^{2} x^{2} + (3 \cdot 4 \cdot - 1) \cdot - 1 x^{3} x^{2} + x^{3} (- - 1) \cdot - 1 d x$

Step 2.35

Move $x^{3}$ .

$64 \int (4 \cdot 4) \cdot 4 x^{3} x^{2} x^{2} x^{2} + (3 \cdot 4 \cdot 4) \cdot - 1 x^{3} x^{2} x^{2} + (3 \cdot 4 \cdot - 1) \cdot - 1 x^{3} x^{2} + (- - 1) \cdot - 1 x^{3} d x$

Step 2.36

Multiply $4$ by $4$ .

$64 \int 16 \cdot 4 x^{3} x^{2} x^{2} x^{2} + (3 \cdot 4 \cdot 4) \cdot - 1 x^{3} x^{2} x^{2} + (3 \cdot 4 \cdot - 1) \cdot - 1 x^{3} x^{2} + (- - 1) \cdot - 1 x^{3} d x$

Step 2.37

Multiply $16$ by $4$ .

$64 \int 64 x^{3} x^{2} x^{2} x^{2} + (3 \cdot 4 \cdot 4) \cdot - 1 x^{3} x^{2} x^{2} + (3 \cdot 4 \cdot - 1) \cdot - 1 x^{3} x^{2} + (- - 1) \cdot - 1 x^{3} d x$

Step 2.38

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

$64 \int 64 x^{3 + 2} x^{2} x^{2} + (3 \cdot 4 \cdot 4) \cdot - 1 x^{3} x^{2} x^{2} + (3 \cdot 4 \cdot - 1) \cdot - 1 x^{3} x^{2} + (- - 1) \cdot - 1 x^{3} d x$

Step 2.39

Add $3$ and $2$ .

$64 \int 64 x^{5} x^{2} x^{2} + (3 \cdot 4 \cdot 4) \cdot - 1 x^{3} x^{2} x^{2} + (3 \cdot 4 \cdot - 1) \cdot - 1 x^{3} x^{2} + (- - 1) \cdot - 1 x^{3} d x$

Step 2.40

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

$64 \int 64 x^{5 + 2} x^{2} + (3 \cdot 4 \cdot 4) \cdot - 1 x^{3} x^{2} x^{2} + (3 \cdot 4 \cdot - 1) \cdot - 1 x^{3} x^{2} + (- - 1) \cdot - 1 x^{3} d x$

Step 2.41

Add $5$ and $2$ .

$64 \int 64 x^{7} x^{2} + (3 \cdot 4 \cdot 4) \cdot - 1 x^{3} x^{2} x^{2} + (3 \cdot 4 \cdot - 1) \cdot - 1 x^{3} x^{2} + (- - 1) \cdot - 1 x^{3} d x$

Step 2.42

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

$64 \int 64 x^{7 + 2} + (3 \cdot 4 \cdot 4) \cdot - 1 x^{3} x^{2} x^{2} + (3 \cdot 4 \cdot - 1) \cdot - 1 x^{3} x^{2} + (- - 1) \cdot - 1 x^{3} d x$

Step 2.43

Add $7$ and $2$ .

$64 \int 64 x^{9} + (3 \cdot 4 \cdot 4) \cdot - 1 x^{3} x^{2} x^{2} + (3 \cdot 4 \cdot - 1) \cdot - 1 x^{3} x^{2} + (- - 1) \cdot - 1 x^{3} d x$

Step 2.44

Multiply $3$ by $4$ .

$64 \int 64 x^{9} + 12 \cdot 4 \cdot - 1 x^{3} x^{2} x^{2} + (3 \cdot 4 \cdot - 1) \cdot - 1 x^{3} x^{2} + (- - 1) \cdot - 1 x^{3} d x$

Step 2.45

Multiply $12$ by $4$ .

$64 \int 64 x^{9} + 48 \cdot - 1 x^{3} x^{2} x^{2} + (3 \cdot 4 \cdot - 1) \cdot - 1 x^{3} x^{2} + (- - 1) \cdot - 1 x^{3} d x$

Step 2.46

Multiply $48$ by $- 1$ .

$64 \int 64 x^{9} - 48 x^{3} x^{2} x^{2} + (3 \cdot 4 \cdot - 1) \cdot - 1 x^{3} x^{2} + (- - 1) \cdot - 1 x^{3} d x$

Step 2.47

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

$64 \int 64 x^{9} - 48 x^{3 + 2} x^{2} + (3 \cdot 4 \cdot - 1) \cdot - 1 x^{3} x^{2} + (- - 1) \cdot - 1 x^{3} d x$

Step 2.48

Add $3$ and $2$ .

$64 \int 64 x^{9} - 48 x^{5} x^{2} + (3 \cdot 4 \cdot - 1) \cdot - 1 x^{3} x^{2} + (- - 1) \cdot - 1 x^{3} d x$

Step 2.49

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

$64 \int 64 x^{9} - 48 x^{5 + 2} + (3 \cdot 4 \cdot - 1) \cdot - 1 x^{3} x^{2} + (- - 1) \cdot - 1 x^{3} d x$

Step 2.50

Add $5$ and $2$ .

$64 \int 64 x^{9} - 48 x^{7} + (3 \cdot 4 \cdot - 1) \cdot - 1 x^{3} x^{2} + (- - 1) \cdot - 1 x^{3} d x$

Step 2.51

Multiply $3$ by $4$ .

$64 \int 64 x^{9} - 48 x^{7} + 12 \cdot - 1 \cdot - 1 x^{3} x^{2} + (- - 1) \cdot - 1 x^{3} d x$

Step 2.52

Multiply $12$ by $- 1$ .

$64 \int 64 x^{9} - 48 x^{7} - 12 \cdot - 1 x^{3} x^{2} + (- - 1) \cdot - 1 x^{3} d x$

Step 2.53

Multiply $- 12$ by $- 1$ .

$64 \int 64 x^{9} - 48 x^{7} + 12 x^{3} x^{2} + (- - 1) \cdot - 1 x^{3} d x$

Step 2.54

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

$64 \int 64 x^{9} - 48 x^{7} + 12 x^{3 + 2} + (- - 1) \cdot - 1 x^{3} d x$

Step 2.55

Add $3$ and $2$ .

$64 \int 64 x^{9} - 48 x^{7} + 12 x^{5} + (- - 1) \cdot - 1 x^{3} d x$

Step 2.56

Multiply $- 1$ by $- 1$ .

$64 \int 64 x^{9} - 48 x^{7} + 12 x^{5} + 1 \cdot - 1 x^{3} d x$

Step 2.57

Multiply $- 1$ by $1$ .

$64 \int 64 x^{9} - 48 x^{7} + 12 x^{5} - x^{3} d x$

Step 3

Split the single integral into multiple integrals.

$64 (\int 64 x^{9} d x + \int - 48 x^{7} d x + \int 12 x^{5} d x + \int - x^{3} d x)$

Step 4

Since $64$ is constant with respect to $x$ , move $64$ out of the integral.

$64 (64 \int x^{9} d x + \int - 48 x^{7} d x + \int 12 x^{5} d x + \int - x^{3} d x)$

Step 5

By the Power Rule, the integral of $x^{9}$ with respect to $x$ is $\frac{1}{10} x^{10}$ .

$64 (64 (\frac{1}{10} x^{10} + C) + \int - 48 x^{7} d x + \int 12 x^{5} d x + \int - x^{3} d x)$

Step 6

Since $- 48$ is constant with respect to $x$ , move $- 48$ out of the integral.

$64 (64 (\frac{1}{10} x^{10} + C) - 48 \int x^{7} d x + \int 12 x^{5} d x + \int - x^{3} d x)$

Step 7

By the Power Rule, the integral of $x^{7}$ with respect to $x$ is $\frac{1}{8} x^{8}$ .

$64 (64 (\frac{1}{10} x^{10} + C) - 48 (\frac{1}{8} x^{8} + C) + \int 12 x^{5} d x + \int - x^{3} d x)$

Step 8

Since $12$ is constant with respect to $x$ , move $12$ out of the integral.

$64 (64 (\frac{1}{10} x^{10} + C) - 48 (\frac{1}{8} x^{8} + C) + 12 \int x^{5} d x + \int - x^{3} d x)$

Step 9

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

$64 (64 (\frac{1}{10} x^{10} + C) - 48 (\frac{1}{8} x^{8} + C) + 12 (\frac{1}{6} x^{6} + C) + \int - x^{3} d x)$

Step 10

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

$64 (64 (\frac{1}{10} x^{10} + C) - 48 (\frac{1}{8} x^{8} + C) + 12 (\frac{1}{6} x^{6} + C) - \int x^{3} d x)$

Step 11

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

$64 (64 (\frac{1}{10} x^{10} + C) - 48 (\frac{1}{8} x^{8} + C) + 12 (\frac{1}{6} x^{6} + C) - (\frac{1}{4} x^{4} + C))$

Step 12

Simplify.

$64 (\frac{32 x^{10}}{5} - 6 x^{8} + 2 x^{6} - \frac{x^{4}}{4}) + C$

Step 13

Reorder terms.

$64 (\frac{32}{5} x^{10} - 6 x^{8} + 2 x^{6} - \frac{1}{4} x^{4}) + C$