Calculus Examples

$\int x (3 x - 4) (4 x + 3) d x$

Step 1

Let $u = 4 x + 3$ . Then $d u = 4 d x$ , so $\frac{1}{4} d u = d x$ . Rewrite using $u$ and $d$ $u$ .

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Step 1.1

Let $u = 4 x + 3$ . Find $\frac{d u}{d x}$ .

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Step 1.1.1

Differentiate $4 x + 3$ .

$\frac{d}{d x} [4 x + 3]$

Step 1.1.2

By the Sum Rule, the derivative of $4 x + 3$ with respect to $x$ is $\frac{d}{d x} [4 x] + \frac{d}{d x} [3]$ .

$\frac{d}{d x} [4 x] + \frac{d}{d x} [3]$

Step 1.1.3

Evaluate $\frac{d}{d x} [4 x]$ .

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Step 1.1.3.1

Since $4$ is constant with respect to $x$ , the derivative of $4 x$ with respect to $x$ is $4 \frac{d}{d x} [x]$ .

$4 \frac{d}{d x} [x] + \frac{d}{d x} [3]$

Step 1.1.3.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 1$ .

$4 \cdot 1 + \frac{d}{d x} [3]$

Step 1.1.3.3

Multiply $4$ by $1$ .

$4 + \frac{d}{d x} [3]$

$4 + \frac{d}{d x} [3]$

Step 1.1.4

Differentiate using the Constant Rule.

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Step 1.1.4.1

Since $3$ is constant with respect to $x$ , the derivative of $3$ with respect to $x$ is $0$ .

$4 + 0$

Step 1.1.4.2

Add $4$ and $0$ .

$4$

$4$

$4$

Step 1.2

Rewrite the problem using $u$ and $d u$ .

$\int (\frac{u}{4} - \frac{3}{4}) (3 (\frac{u}{4} - \frac{3}{4}) - 4) u \frac{1}{4} d u$

$\int (\frac{u}{4} - \frac{3}{4}) (3 (\frac{u}{4} - \frac{3}{4}) - 4) u \frac{1}{4} d u$

Step 2

Combine $\frac{1}{4}$ and $u$ .

$\int (\frac{u}{4} - \frac{3}{4}) (3 (\frac{u}{4} - \frac{3}{4}) - 4) \frac{u}{4} d u$

Step 3

Simplify.

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Step 3.1

Apply the distributive property.

$\int (\frac{u}{4} - \frac{3}{4}) (3 \frac{u}{4} + 3 (- \frac{3}{4}) - 4) \frac{u}{4} d u$

Step 3.2

Apply the distributive property.

$\int (\frac{u}{4} (3 \frac{u}{4} + 3 (- \frac{3}{4}) - 4) - \frac{3}{4} (3 \frac{u}{4} + 3 (- \frac{3}{4}) - 4)) \frac{u}{4} d u$

Step 3.3

Apply the distributive property.

$\int (\frac{u}{4} (3 \frac{u}{4} + 3 (- \frac{3}{4})) + \frac{u}{4} \cdot - 4 - \frac{3}{4} (3 \frac{u}{4} + 3 (- \frac{3}{4}) - 4)) \frac{u}{4} d u$

Step 3.4

Apply the distributive property.

$\int (\frac{u}{4} (3 \frac{u}{4}) + \frac{u}{4} (3 (- \frac{3}{4})) + \frac{u}{4} \cdot - 4 - \frac{3}{4} (3 \frac{u}{4} + 3 (- \frac{3}{4}) - 4)) \frac{u}{4} d u$

Step 3.5

Apply the distributive property.

$\int (\frac{u}{4} (3 \frac{u}{4}) + \frac{u}{4} (3 (- \frac{3}{4})) + \frac{u}{4} \cdot - 4 - \frac{3}{4} (3 \frac{u}{4} + 3 (- \frac{3}{4})) - \frac{3}{4} \cdot - 4) \frac{u}{4} d u$

Step 3.6

Apply the distributive property.

$\int (\frac{u}{4} (3 \frac{u}{4}) + \frac{u}{4} (3 (- \frac{3}{4})) + \frac{u}{4} \cdot - 4 - \frac{3}{4} (3 \frac{u}{4}) - \frac{3}{4} (3 (- \frac{3}{4})) - \frac{3}{4} \cdot - 4) \frac{u}{4} d u$

Step 3.7

Apply the distributive property.

$\int (\frac{u}{4} (3 \frac{u}{4}) + \frac{u}{4} (3 (- \frac{3}{4})) + \frac{u}{4} \cdot - 4) \frac{u}{4} + (- \frac{3}{4} (3 \frac{u}{4}) - \frac{3}{4} (3 (- \frac{3}{4})) - \frac{3}{4} \cdot - 4) \frac{u}{4} d u$

Step 3.8

Apply the distributive property.

$\int (\frac{u}{4} (3 \frac{u}{4}) + \frac{u}{4} (3 (- \frac{3}{4}))) \frac{u}{4} + \frac{u}{4} \cdot - 4 \frac{u}{4} + (- \frac{3}{4} (3 \frac{u}{4}) - \frac{3}{4} (3 (- \frac{3}{4})) - \frac{3}{4} \cdot - 4) \frac{u}{4} d u$

Step 3.9

Apply the distributive property.

$\int \frac{u}{4} (3 \frac{u}{4}) \frac{u}{4} + \frac{u}{4} (3 (- \frac{3}{4})) \frac{u}{4} + \frac{u}{4} \cdot - 4 \frac{u}{4} + (- \frac{3}{4} (3 \frac{u}{4}) - \frac{3}{4} (3 (- \frac{3}{4})) - \frac{3}{4} \cdot - 4) \frac{u}{4} d u$

Step 3.10

Apply the distributive property.

$\int \frac{u}{4} (3 \frac{u}{4}) \frac{u}{4} + \frac{u}{4} (3 (- \frac{3}{4})) \frac{u}{4} + \frac{u}{4} \cdot - 4 \frac{u}{4} + (- \frac{3}{4} (3 \frac{u}{4}) - \frac{3}{4} (3 (- \frac{3}{4}))) \frac{u}{4} - \frac{3}{4} \cdot - 4 \frac{u}{4} d u$

Step 3.11

Apply the distributive property.

$\int \frac{u}{4} (3 \frac{u}{4}) \frac{u}{4} + \frac{u}{4} (3 (- \frac{3}{4})) \frac{u}{4} + \frac{u}{4} \cdot - 4 \frac{u}{4} - \frac{3}{4} (3 \frac{u}{4}) \frac{u}{4} - \frac{3}{4} (3 (- \frac{3}{4})) \frac{u}{4} - \frac{3}{4} \cdot - 4 \frac{u}{4} d u$

Step 3.12

Reorder $\frac{u}{4}$ and $3$ .

$\int 3 \cdot \frac{u}{4} \frac{u}{4} \frac{u}{4} + \frac{u}{4} (3 (- \frac{3}{4})) \frac{u}{4} + \frac{u}{4} \cdot - 4 \frac{u}{4} - \frac{3}{4} (3 \frac{u}{4}) \frac{u}{4} - \frac{3}{4} (3 (- \frac{3}{4})) \frac{u}{4} - \frac{3}{4} \cdot - 4 \frac{u}{4} d u$

Step 3.13

Move parentheses.

$\int 3 \cdot \frac{u}{4} \frac{u}{4} \frac{u}{4} + \frac{u}{4} (3 \cdot - 1) \frac{3}{4} \frac{u}{4} + \frac{u}{4} \cdot - 4 \frac{u}{4} - \frac{3}{4} (3 \frac{u}{4}) \frac{u}{4} - \frac{3}{4} (3 (- \frac{3}{4})) \frac{u}{4} - \frac{3}{4} \cdot - 4 \frac{u}{4} d u$

Step 3.14

Move $\frac{u}{4}$ .

$\int 3 \cdot \frac{u}{4} \frac{u}{4} \frac{u}{4} + 3 \cdot - 1 \frac{u}{4} \frac{3}{4} \frac{u}{4} + \frac{u}{4} \cdot - 4 \frac{u}{4} - \frac{3}{4} (3 \frac{u}{4}) \frac{u}{4} - \frac{3}{4} (3 (- \frac{3}{4})) \frac{u}{4} - \frac{3}{4} \cdot - 4 \frac{u}{4} d u$

Step 3.15

Reorder $\frac{u}{4}$ and $- 4$ .

$\int 3 \cdot \frac{u}{4} \frac{u}{4} \frac{u}{4} + 3 \cdot - 1 \frac{u}{4} \frac{3}{4} \frac{u}{4} - 4 \cdot \frac{u}{4} \frac{u}{4} - \frac{3}{4} (3 \frac{u}{4}) \frac{u}{4} - \frac{3}{4} (3 (- \frac{3}{4})) \frac{u}{4} - \frac{3}{4} \cdot - 4 \frac{u}{4} d u$

Step 3.16

Move $\frac{3}{4}$ .

$\int 3 \cdot \frac{u}{4} \frac{u}{4} \frac{u}{4} + 3 \cdot - 1 \frac{u}{4} \frac{3}{4} \frac{u}{4} - 4 \cdot \frac{u}{4} \frac{u}{4} - 1 \cdot 3 \frac{3}{4} \frac{u}{4} \frac{u}{4} - \frac{3}{4} (3 (- \frac{3}{4})) \frac{u}{4} - \frac{3}{4} \cdot - 4 \frac{u}{4} d u$

Step 3.17

Move parentheses.

$\int 3 \cdot \frac{u}{4} \frac{u}{4} \frac{u}{4} + 3 \cdot - 1 \frac{u}{4} \frac{3}{4} \frac{u}{4} - 4 \cdot \frac{u}{4} \frac{u}{4} - 1 \cdot 3 \frac{3}{4} \frac{u}{4} \frac{u}{4} - \frac{3}{4} (3 \cdot - 1) \frac{3}{4} \frac{u}{4} - \frac{3}{4} \cdot - 4 \frac{u}{4} d u$

Step 3.18

Move $\frac{3}{4}$ .

$\int 3 \cdot \frac{u}{4} \frac{u}{4} \frac{u}{4} + 3 \cdot - 1 \frac{u}{4} \frac{3}{4} \frac{u}{4} - 4 \cdot \frac{u}{4} \frac{u}{4} - 1 \cdot 3 \frac{3}{4} \frac{u}{4} \frac{u}{4} - 1 \cdot 3 \cdot - 1 \frac{3}{4} \frac{3}{4} \frac{u}{4} - \frac{3}{4} \cdot - 4 \frac{u}{4} d u$

Step 3.19

Move $\frac{3}{4}$ .

$\int 3 \cdot \frac{u}{4} \frac{u}{4} \frac{u}{4} + 3 \cdot - 1 \frac{u}{4} \frac{3}{4} \frac{u}{4} - 4 \cdot \frac{u}{4} \frac{u}{4} - 1 \cdot 3 \frac{3}{4} \frac{u}{4} \frac{u}{4} - 1 \cdot 3 \cdot - 1 \frac{3}{4} \frac{3}{4} \frac{u}{4} - 1 \cdot - 4 \frac{3}{4} \frac{u}{4} d u$

Step 3.20

Combine $3$ and $\frac{u}{4}$ .

$\int \frac{3 u}{4} \cdot \frac{u}{4} \frac{u}{4} + 3 \cdot - 1 \frac{u}{4} \frac{3}{4} \frac{u}{4} - 4 \frac{u}{4} \frac{u}{4} - 1 \cdot 3 \frac{3}{4} \frac{u}{4} \frac{u}{4} - 1 \cdot 3 \cdot - 1 \frac{3}{4} \frac{3}{4} \frac{u}{4} - 1 \cdot - 4 \frac{3}{4} \frac{u}{4} d u$

Step 3.21

Multiply $\frac{3 u}{4}$ by $\frac{u}{4}$ .

$\int \frac{3 u \cdot u}{4 \cdot 4} \cdot \frac{u}{4} + 3 \cdot - 1 \frac{u}{4} \frac{3}{4} \frac{u}{4} - 4 \frac{u}{4} \frac{u}{4} - 1 \cdot 3 \frac{3}{4} \frac{u}{4} \frac{u}{4} - 1 \cdot 3 \cdot - 1 \frac{3}{4} \frac{3}{4} \frac{u}{4} - 1 \cdot - 4 \frac{3}{4} \frac{u}{4} d u$

Step 3.22

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

$\int \frac{3 (u^{1} u)}{4 \cdot 4} \cdot \frac{u}{4} + 3 \cdot - 1 \frac{u}{4} \frac{3}{4} \frac{u}{4} - 4 \frac{u}{4} \frac{u}{4} - 1 \cdot 3 \frac{3}{4} \frac{u}{4} \frac{u}{4} - 1 \cdot 3 \cdot - 1 \frac{3}{4} \frac{3}{4} \frac{u}{4} - 1 \cdot - 4 \frac{3}{4} \frac{u}{4} d u$

Step 3.23

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

$\int \frac{3 (u^{1} u^{1})}{4 \cdot 4} \cdot \frac{u}{4} + 3 \cdot - 1 \frac{u}{4} \frac{3}{4} \frac{u}{4} - 4 \frac{u}{4} \frac{u}{4} - 1 \cdot 3 \frac{3}{4} \frac{u}{4} \frac{u}{4} - 1 \cdot 3 \cdot - 1 \frac{3}{4} \frac{3}{4} \frac{u}{4} - 1 \cdot - 4 \frac{3}{4} \frac{u}{4} d u$

Step 3.24

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

$\int \frac{3 u^{1 + 1}}{4 \cdot 4} \cdot \frac{u}{4} + 3 \cdot - 1 \frac{u}{4} \frac{3}{4} \frac{u}{4} - 4 \frac{u}{4} \frac{u}{4} - 1 \cdot 3 \frac{3}{4} \frac{u}{4} \frac{u}{4} - 1 \cdot 3 \cdot - 1 \frac{3}{4} \frac{3}{4} \frac{u}{4} - 1 \cdot - 4 \frac{3}{4} \frac{u}{4} d u$

Step 3.25

Add $1$ and $1$ .

$\int \frac{3 u^{2}}{4 \cdot 4} \cdot \frac{u}{4} + 3 \cdot - 1 \frac{u}{4} \frac{3}{4} \frac{u}{4} - 4 \frac{u}{4} \frac{u}{4} - 1 \cdot 3 \frac{3}{4} \frac{u}{4} \frac{u}{4} - 1 \cdot 3 \cdot - 1 \frac{3}{4} \frac{3}{4} \frac{u}{4} - 1 \cdot - 4 \frac{3}{4} \frac{u}{4} d u$

Step 3.26

Multiply $4$ by $4$ .

$\int \frac{3 u^{2}}{16} \cdot \frac{u}{4} + 3 \cdot - 1 \frac{u}{4} \frac{3}{4} \frac{u}{4} - 4 \frac{u}{4} \frac{u}{4} - 1 \cdot 3 \frac{3}{4} \frac{u}{4} \frac{u}{4} - 1 \cdot 3 \cdot - 1 \frac{3}{4} \frac{3}{4} \frac{u}{4} - 1 \cdot - 4 \frac{3}{4} \frac{u}{4} d u$

Step 3.27

Multiply $\frac{3 u^{2}}{16}$ by $\frac{u}{4}$ .

$\int \frac{3 u^{2} u}{16 \cdot 4} + 3 \cdot - 1 \frac{u}{4} \frac{3}{4} \frac{u}{4} - 4 \frac{u}{4} \frac{u}{4} - 1 \cdot 3 \frac{3}{4} \frac{u}{4} \frac{u}{4} - 1 \cdot 3 \cdot - 1 \frac{3}{4} \frac{3}{4} \frac{u}{4} - 1 \cdot - 4 \frac{3}{4} \frac{u}{4} d u$

Step 3.28

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

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

Step 3.29

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

$\int \frac{3 u^{2 + 1}}{16 \cdot 4} + 3 \cdot - 1 \frac{u}{4} \frac{3}{4} \frac{u}{4} - 4 \frac{u}{4} \frac{u}{4} - 1 \cdot 3 \frac{3}{4} \frac{u}{4} \frac{u}{4} - 1 \cdot 3 \cdot - 1 \frac{3}{4} \frac{3}{4} \frac{u}{4} - 1 \cdot - 4 \frac{3}{4} \frac{u}{4} d u$

Step 3.30

Add $2$ and $1$ .

$\int \frac{3 u^{3}}{16 \cdot 4} + 3 \cdot - 1 \frac{u}{4} \frac{3}{4} \frac{u}{4} - 4 \frac{u}{4} \frac{u}{4} - 1 \cdot 3 \frac{3}{4} \frac{u}{4} \frac{u}{4} - 1 \cdot 3 \cdot - 1 \frac{3}{4} \frac{3}{4} \frac{u}{4} - 1 \cdot - 4 \frac{3}{4} \frac{u}{4} d u$

Step 3.31

Multiply $16$ by $4$ .

$\int \frac{3 u^{3}}{64} + 3 \cdot - 1 \frac{u}{4} \frac{3}{4} \frac{u}{4} - 4 \frac{u}{4} \frac{u}{4} - 1 \cdot 3 \frac{3}{4} \frac{u}{4} \frac{u}{4} - 1 \cdot 3 \cdot - 1 \frac{3}{4} \frac{3}{4} \frac{u}{4} - 1 \cdot - 4 \frac{3}{4} \frac{u}{4} d u$

Step 3.32

Multiply $3$ by $- 1$ .

$\int \frac{3 u^{3}}{64} - 3 \frac{u}{4} \frac{3}{4} \frac{u}{4} - 4 \frac{u}{4} \frac{u}{4} - 1 \cdot 3 \frac{3}{4} \frac{u}{4} \frac{u}{4} - 1 \cdot 3 \cdot - 1 \frac{3}{4} \frac{3}{4} \frac{u}{4} - 1 \cdot - 4 \frac{3}{4} \frac{u}{4} d u$

Step 3.33

Combine $- 3$ and $\frac{u}{4}$ .

$\int \frac{3 u^{3}}{64} + \frac{- 3 u}{4} \cdot \frac{3}{4} \frac{u}{4} - 4 \frac{u}{4} \frac{u}{4} - 1 \cdot 3 \frac{3}{4} \frac{u}{4} \frac{u}{4} - 1 \cdot 3 \cdot - 1 \frac{3}{4} \frac{3}{4} \frac{u}{4} - 1 \cdot - 4 \frac{3}{4} \frac{u}{4} d u$

Step 3.34

Multiply $\frac{- 3 u}{4}$ by $\frac{3}{4}$ .

$\int \frac{3 u^{3}}{64} + \frac{- 3 u \cdot 3}{4 \cdot 4} \cdot \frac{u}{4} - 4 \frac{u}{4} \frac{u}{4} - 1 \cdot 3 \frac{3}{4} \frac{u}{4} \frac{u}{4} - 1 \cdot 3 \cdot - 1 \frac{3}{4} \frac{3}{4} \frac{u}{4} - 1 \cdot - 4 \frac{3}{4} \frac{u}{4} d u$

Step 3.35

Multiply $4$ by $4$ .

$\int \frac{3 u^{3}}{64} + \frac{- 3 u \cdot 3}{16} \cdot \frac{u}{4} - 4 \frac{u}{4} \frac{u}{4} - 1 \cdot 3 \frac{3}{4} \frac{u}{4} \frac{u}{4} - 1 \cdot 3 \cdot - 1 \frac{3}{4} \frac{3}{4} \frac{u}{4} - 1 \cdot - 4 \frac{3}{4} \frac{u}{4} d u$

Step 3.36

Multiply $\frac{- 3 u \cdot 3}{16}$ by $\frac{u}{4}$ .

$\int \frac{3 u^{3}}{64} + \frac{- 3 u \cdot 3 u}{16 \cdot 4} - 4 \frac{u}{4} \frac{u}{4} - 1 \cdot 3 \frac{3}{4} \frac{u}{4} \frac{u}{4} - 1 \cdot 3 \cdot - 1 \frac{3}{4} \frac{3}{4} \frac{u}{4} - 1 \cdot - 4 \frac{3}{4} \frac{u}{4} d u$

Step 3.37

Multiply $16$ by $4$ .

$\int \frac{3 u^{3}}{64} + \frac{- 3 u \cdot 3 u}{64} - 4 \frac{u}{4} \frac{u}{4} - 1 \cdot 3 \frac{3}{4} \frac{u}{4} \frac{u}{4} - 1 \cdot 3 \cdot - 1 \frac{3}{4} \frac{3}{4} \frac{u}{4} - 1 \cdot - 4 \frac{3}{4} \frac{u}{4} d u$

Step 3.38

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

$\int \frac{3 u^{3}}{64} + \frac{- 3 u \cdot 3 u}{64} + \frac{- 4 u}{4} \cdot \frac{u}{4} - 1 \cdot 3 \frac{3}{4} \frac{u}{4} \frac{u}{4} - 1 \cdot 3 \cdot - 1 \frac{3}{4} \frac{3}{4} \frac{u}{4} - 1 \cdot - 4 \frac{3}{4} \frac{u}{4} d u$

Step 3.39

Multiply $\frac{- 4 u}{4}$ by $\frac{u}{4}$ .

$\int \frac{3 u^{3}}{64} + \frac{- 3 u \cdot 3 u}{64} + \frac{- 4 u \cdot u}{4 \cdot 4} - 1 \cdot 3 \frac{3}{4} \frac{u}{4} \frac{u}{4} - 1 \cdot 3 \cdot - 1 \frac{3}{4} \frac{3}{4} \frac{u}{4} - 1 \cdot - 4 \frac{3}{4} \frac{u}{4} d u$

Step 3.40

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

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

Step 3.41

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

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

Step 3.42

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

$\int \frac{3 u^{3}}{64} + \frac{- 3 u \cdot 3 u}{64} + \frac{- 4 u^{1 + 1}}{4 \cdot 4} - 1 \cdot 3 \frac{3}{4} \frac{u}{4} \frac{u}{4} - 1 \cdot 3 \cdot - 1 \frac{3}{4} \frac{3}{4} \frac{u}{4} - 1 \cdot - 4 \frac{3}{4} \frac{u}{4} d u$

Step 3.43

Add $1$ and $1$ .

$\int \frac{3 u^{3}}{64} + \frac{- 3 u \cdot 3 u}{64} + \frac{- 4 u^{2}}{4 \cdot 4} - 1 \cdot 3 \frac{3}{4} \frac{u}{4} \frac{u}{4} - 1 \cdot 3 \cdot - 1 \frac{3}{4} \frac{3}{4} \frac{u}{4} - 1 \cdot - 4 \frac{3}{4} \frac{u}{4} d u$

Step 3.44

Multiply $4$ by $4$ .

$\int \frac{3 u^{3}}{64} + \frac{- 3 u \cdot 3 u}{64} + \frac{- 4 u^{2}}{16} - 1 \cdot 3 \frac{3}{4} \frac{u}{4} \frac{u}{4} - 1 \cdot 3 \cdot - 1 \frac{3}{4} \frac{3}{4} \frac{u}{4} - 1 \cdot - 4 \frac{3}{4} \frac{u}{4} d u$

Step 3.45

Multiply $- 1$ by $3$ .

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

Step 3.46

Combine $- 3$ and $\frac{3}{4}$ .

$\int \frac{3 u^{3}}{64} + \frac{- 3 u \cdot 3 u}{64} + \frac{- 4 u^{2}}{16} + \frac{- 3 \cdot 3}{4} \cdot \frac{u}{4} \frac{u}{4} - 1 \cdot 3 \cdot - 1 \frac{3}{4} \frac{3}{4} \frac{u}{4} - 1 \cdot - 4 \frac{3}{4} \frac{u}{4} d u$

Step 3.47

Multiply $- 3$ by $3$ .

$\int \frac{3 u^{3}}{64} + \frac{- 3 u \cdot 3 u}{64} + \frac{- 4 u^{2}}{16} + \frac{- 9}{4} \cdot \frac{u}{4} \frac{u}{4} - 1 \cdot 3 \cdot - 1 \frac{3}{4} \frac{3}{4} \frac{u}{4} - 1 \cdot - 4 \frac{3}{4} \frac{u}{4} d u$

Step 3.48

Multiply $\frac{- 9}{4}$ by $\frac{u}{4}$ .

$\int \frac{3 u^{3}}{64} + \frac{- 3 u \cdot 3 u}{64} + \frac{- 4 u^{2}}{16} + \frac{- 9 u}{4 \cdot 4} \cdot \frac{u}{4} - 1 \cdot 3 \cdot - 1 \frac{3}{4} \frac{3}{4} \frac{u}{4} - 1 \cdot - 4 \frac{3}{4} \frac{u}{4} d u$

Step 3.49

Multiply $4$ by $4$ .

$\int \frac{3 u^{3}}{64} + \frac{- 3 u \cdot 3 u}{64} + \frac{- 4 u^{2}}{16} + \frac{- 9 u}{16} \cdot \frac{u}{4} - 1 \cdot 3 \cdot - 1 \frac{3}{4} \frac{3}{4} \frac{u}{4} - 1 \cdot - 4 \frac{3}{4} \frac{u}{4} d u$

Step 3.50

Multiply $\frac{- 9 u}{16}$ by $\frac{u}{4}$ .

$\int \frac{3 u^{3}}{64} + \frac{- 3 u \cdot 3 u}{64} + \frac{- 4 u^{2}}{16} + \frac{- 9 u \cdot u}{16 \cdot 4} - 1 \cdot 3 \cdot - 1 \frac{3}{4} \frac{3}{4} \frac{u}{4} - 1 \cdot - 4 \frac{3}{4} \frac{u}{4} d u$

Step 3.51

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

$\int \frac{3 u^{3}}{64} + \frac{- 3 u \cdot 3 u}{64} + \frac{- 4 u^{2}}{16} + \frac{- 9 (u^{1} u)}{16 \cdot 4} - 1 \cdot 3 \cdot - 1 \frac{3}{4} \frac{3}{4} \frac{u}{4} - 1 \cdot - 4 \frac{3}{4} \frac{u}{4} d u$

Step 3.52

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

$\int \frac{3 u^{3}}{64} + \frac{- 3 u \cdot 3 u}{64} + \frac{- 4 u^{2}}{16} + \frac{- 9 (u^{1} u^{1})}{16 \cdot 4} - 1 \cdot 3 \cdot - 1 \frac{3}{4} \frac{3}{4} \frac{u}{4} - 1 \cdot - 4 \frac{3}{4} \frac{u}{4} d u$

Step 3.53

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

$\int \frac{3 u^{3}}{64} + \frac{- 3 u \cdot 3 u}{64} + \frac{- 4 u^{2}}{16} + \frac{- 9 u^{1 + 1}}{16 \cdot 4} - 1 \cdot 3 \cdot - 1 \frac{3}{4} \frac{3}{4} \frac{u}{4} - 1 \cdot - 4 \frac{3}{4} \frac{u}{4} d u$

Step 3.54

Add $1$ and $1$ .

$\int \frac{3 u^{3}}{64} + \frac{- 3 u \cdot 3 u}{64} + \frac{- 4 u^{2}}{16} + \frac{- 9 u^{2}}{16 \cdot 4} - 1 \cdot 3 \cdot - 1 \frac{3}{4} \frac{3}{4} \frac{u}{4} - 1 \cdot - 4 \frac{3}{4} \frac{u}{4} d u$

Step 3.55

Multiply $16$ by $4$ .

$\int \frac{3 u^{3}}{64} + \frac{- 3 u \cdot 3 u}{64} + \frac{- 4 u^{2}}{16} + \frac{- 9 u^{2}}{64} - 1 \cdot 3 \cdot - 1 \frac{3}{4} \frac{3}{4} \frac{u}{4} - 1 \cdot - 4 \frac{3}{4} \frac{u}{4} d u$

Step 3.56

Multiply $- 1$ by $3$ .

$\int \frac{3 u^{3}}{64} + \frac{- 3 u \cdot 3 u}{64} + \frac{- 4 u^{2}}{16} + \frac{- 9 u^{2}}{64} - 3 \cdot - 1 \frac{3}{4} \frac{3}{4} \frac{u}{4} - 1 \cdot - 4 \frac{3}{4} \frac{u}{4} d u$

Step 3.57

Multiply $- 3$ by $- 1$ .

$\int \frac{3 u^{3}}{64} + \frac{- 3 u \cdot 3 u}{64} + \frac{- 4 u^{2}}{16} + \frac{- 9 u^{2}}{64} + 3 (\frac{3}{4}) \frac{3}{4} \frac{u}{4} - 1 \cdot - 4 \frac{3}{4} \frac{u}{4} d u$

Step 3.58

Combine $3$ and $\frac{3}{4}$ .

$\int \frac{3 u^{3}}{64} + \frac{- 3 u \cdot 3 u}{64} + \frac{- 4 u^{2}}{16} + \frac{- 9 u^{2}}{64} + \frac{3 \cdot 3}{4} \cdot \frac{3}{4} \frac{u}{4} - 1 \cdot - 4 \frac{3}{4} \frac{u}{4} d u$

Step 3.59

Multiply $3$ by $3$ .

$\int \frac{3 u^{3}}{64} + \frac{- 3 u \cdot 3 u}{64} + \frac{- 4 u^{2}}{16} + \frac{- 9 u^{2}}{64} + \frac{9}{4} \cdot \frac{3}{4} \frac{u}{4} - 1 \cdot - 4 \frac{3}{4} \frac{u}{4} d u$

Step 3.60

Multiply $\frac{9}{4}$ by $\frac{3}{4}$ .

$\int \frac{3 u^{3}}{64} + \frac{- 3 u \cdot 3 u}{64} + \frac{- 4 u^{2}}{16} + \frac{- 9 u^{2}}{64} + \frac{9 \cdot 3}{4 \cdot 4} \cdot \frac{u}{4} - 1 \cdot - 4 \frac{3}{4} \frac{u}{4} d u$

Step 3.61

Multiply $9$ by $3$ .

$\int \frac{3 u^{3}}{64} + \frac{- 3 u \cdot 3 u}{64} + \frac{- 4 u^{2}}{16} + \frac{- 9 u^{2}}{64} + \frac{27}{4 \cdot 4} \cdot \frac{u}{4} - 1 \cdot - 4 \frac{3}{4} \frac{u}{4} d u$

Step 3.62

Multiply $4$ by $4$ .

$\int \frac{3 u^{3}}{64} + \frac{- 3 u \cdot 3 u}{64} + \frac{- 4 u^{2}}{16} + \frac{- 9 u^{2}}{64} + \frac{27}{16} \cdot \frac{u}{4} - 1 \cdot - 4 \frac{3}{4} \frac{u}{4} d u$

Step 3.63

Multiply $\frac{27}{16}$ by $\frac{u}{4}$ .

$\int \frac{3 u^{3}}{64} + \frac{- 3 u \cdot 3 u}{64} + \frac{- 4 u^{2}}{16} + \frac{- 9 u^{2}}{64} + \frac{27 u}{16 \cdot 4} - 1 \cdot - 4 \frac{3}{4} \frac{u}{4} d u$

Step 3.64

Multiply $16$ by $4$ .

$\int \frac{3 u^{3}}{64} + \frac{- 3 u \cdot 3 u}{64} + \frac{- 4 u^{2}}{16} + \frac{- 9 u^{2}}{64} + \frac{27 u}{64} - 1 \cdot - 4 \frac{3}{4} \frac{u}{4} d u$

Step 3.65

Multiply $- 1$ by $- 4$ .

$\int \frac{3 u^{3}}{64} + \frac{- 3 u \cdot 3 u}{64} + \frac{- 4 u^{2}}{16} + \frac{- 9 u^{2}}{64} + \frac{27 u}{64} + 4 (\frac{3}{4}) \frac{u}{4} d u$

Step 3.66

Combine $4$ and $\frac{3}{4}$ .

$\int \frac{3 u^{3}}{64} + \frac{- 3 u \cdot 3 u}{64} + \frac{- 4 u^{2}}{16} + \frac{- 9 u^{2}}{64} + \frac{27 u}{64} + \frac{4 \cdot 3}{4} \cdot \frac{u}{4} d u$

Step 3.67

Multiply $4$ by $3$ .

$\int \frac{3 u^{3}}{64} + \frac{- 3 u \cdot 3 u}{64} + \frac{- 4 u^{2}}{16} + \frac{- 9 u^{2}}{64} + \frac{27 u}{64} + \frac{12}{4} \cdot \frac{u}{4} d u$

Step 3.68

Multiply $\frac{12}{4}$ by $\frac{u}{4}$ .

$\int \frac{3 u^{3}}{64} + \frac{- 3 u \cdot 3 u}{64} + \frac{- 4 u^{2}}{16} + \frac{- 9 u^{2}}{64} + \frac{27 u}{64} + \frac{12 u}{4 \cdot 4} d u$

Step 3.69

Multiply $4$ by $4$ .

$\int \frac{3 u^{3}}{64} + \frac{- 3 u \cdot 3 u}{64} + \frac{- 4 u^{2}}{16} + \frac{- 9 u^{2}}{64} + \frac{27 u}{64} + \frac{12 u}{16} d u$

Step 3.70

To write $\frac{12 u}{16}$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$\int \frac{3 u^{3}}{64} + \frac{- 3 u \cdot 3 u}{64} + \frac{- 4 u^{2}}{16} + \frac{- 9 u^{2}}{64} + \frac{27 u}{64} + \frac{12 u}{16} \cdot \frac{4}{4} d u$

Step 3.71

Write each expression with a common denominator of $64$ , by multiplying each by an appropriate factor of $1$ .

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Step 3.71.1

Multiply $\frac{12 u}{16}$ by $\frac{4}{4}$ .

$\int \frac{3 u^{3}}{64} + \frac{- 3 u \cdot 3 u}{64} + \frac{- 4 u^{2}}{16} + \frac{- 9 u^{2}}{64} + \frac{27 u}{64} + \frac{12 u \cdot 4}{16 \cdot 4} d u$

Step 3.71.2

Multiply $16$ by $4$ .

$\int \frac{3 u^{3}}{64} + \frac{- 3 u \cdot 3 u}{64} + \frac{- 4 u^{2}}{16} + \frac{- 9 u^{2}}{64} + \frac{27 u}{64} + \frac{12 u \cdot 4}{64} d u$

$\int \frac{3 u^{3}}{64} + \frac{- 3 u \cdot 3 u}{64} + \frac{- 4 u^{2}}{16} + \frac{- 9 u^{2}}{64} + \frac{27 u}{64} + \frac{12 u \cdot 4}{64} d u$

Step 3.72

Combine the numerators over the common denominator.

$\int \frac{3 u^{3}}{64} + \frac{- 3 u \cdot 3 u}{64} + \frac{- 4 u^{2}}{16} + \frac{- 9 u^{2}}{64} + \frac{27 u + 12 u \cdot 4}{64} d u$

Step 3.73

Combine the numerators over the common denominator.

$\int \frac{3 u^{3}}{64} + \frac{- 3 u \cdot 3 u}{64} + \frac{- 4 u^{2}}{16} + \frac{- 9 u^{2} + 27 u + 12 u \cdot 4}{64} d u$

Step 3.74

To write $\frac{- 4 u^{2}}{16}$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$\int \frac{3 u^{3}}{64} + \frac{- 3 u \cdot 3 u}{64} + \frac{- 4 u^{2}}{16} \cdot \frac{4}{4} + \frac{- 9 u^{2} + 27 u + 12 u \cdot 4}{64} d u$

Step 3.75

Write each expression with a common denominator of $64$ , by multiplying each by an appropriate factor of $1$ .

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Step 3.75.1

Multiply $\frac{- 4 u^{2}}{16}$ by $\frac{4}{4}$ .

$\int \frac{3 u^{3}}{64} + \frac{- 3 u \cdot 3 u}{64} + \frac{- 4 u^{2} \cdot 4}{16 \cdot 4} + \frac{- 9 u^{2} + 27 u + 12 u \cdot 4}{64} d u$

Step 3.75.2

Multiply $16$ by $4$ .

$\int \frac{3 u^{3}}{64} + \frac{- 3 u \cdot 3 u}{64} + \frac{- 4 u^{2} \cdot 4}{64} + \frac{- 9 u^{2} + 27 u + 12 u \cdot 4}{64} d u$

$\int \frac{3 u^{3}}{64} + \frac{- 3 u \cdot 3 u}{64} + \frac{- 4 u^{2} \cdot 4}{64} + \frac{- 9 u^{2} + 27 u + 12 u \cdot 4}{64} d u$

Step 3.76

Combine the numerators over the common denominator.

$\int \frac{3 u^{3}}{64} + \frac{- 3 u \cdot 3 u}{64} + \frac{- 4 u^{2} \cdot 4 - 9 u^{2} + 27 u + 12 u \cdot 4}{64} d u$

Step 3.77

Combine the numerators over the common denominator.

$\int \frac{3 u^{3}}{64} + \frac{- 3 u \cdot 3 u - 4 u^{2} \cdot 4 - 9 u^{2} + 27 u + 12 u \cdot 4}{64} d u$

Step 3.78

Combine the numerators over the common denominator.

$\int \frac{3 u^{3} - 3 u \cdot 3 u - 4 u^{2} \cdot 4 - 9 u^{2} + 27 u + 12 u \cdot 4}{64} d u$

Step 3.79

Reorder $27 u$ and $12 u \cdot 4$ .

$\int \frac{3 u^{3} - 3 u \cdot 3 u - 4 u^{2} \cdot 4 - 9 u^{2} + 12 u \cdot 4 + 27 u}{64} d u$

Step 3.80

Reorder $- 4 u^{2} \cdot 4$ and $- 9 u^{2}$ .

$\int \frac{3 u^{3} - 3 u \cdot 3 u - 9 u^{2} - 4 u^{2} \cdot 4 + 12 u \cdot 4 + 27 u}{64} d u$

Step 3.81

Reorder $- 3 u \cdot 3 u$ and $- 9 u^{2}$ .

$\int \frac{3 u^{3} - 9 u^{2} - 3 u \cdot 3 u - 4 u^{2} \cdot 4 + 12 u \cdot 4 + 27 u}{64} d u$

$\int \frac{3 u^{3} - 9 u^{2} - 3 u \cdot 3 u - 4 u^{2} \cdot 4 + 12 u \cdot 4 + 27 u}{64} d u$

Step 4

Simplify.

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Step 4.1

Multiply $3$ by $- 3$ .

$\int \frac{3 u^{3} - 9 u^{2} - 9 u \cdot u - 4 u^{2} \cdot 4 + 12 u \cdot 4 + 27 u}{64} d u$

Step 4.2

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

$\int \frac{3 u^{3} - 9 u^{2} - 9 (u^{1} u) - 4 u^{2} \cdot 4 + 12 u \cdot 4 + 27 u}{64} d u$

Step 4.3

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

$\int \frac{3 u^{3} - 9 u^{2} - 9 (u^{1} u^{1}) - 4 u^{2} \cdot 4 + 12 u \cdot 4 + 27 u}{64} d u$

Step 4.4

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

$\int \frac{3 u^{3} - 9 u^{2} - 9 u^{1 + 1} - 4 u^{2} \cdot 4 + 12 u \cdot 4 + 27 u}{64} d u$

Step 4.5

Add $1$ and $1$ .

$\int \frac{3 u^{3} - 9 u^{2} - 9 u^{2} - 4 u^{2} \cdot 4 + 12 u \cdot 4 + 27 u}{64} d u$

Step 4.6

Subtract $9 u^{2}$ from $- 9 u^{2}$ .

$\int \frac{3 u^{3} - 18 u^{2} - 4 u^{2} \cdot 4 + 12 u \cdot 4 + 27 u}{64} d u$

Step 4.7

Multiply $4$ by $- 4$ .

$\int \frac{3 u^{3} - 18 u^{2} - 16 u^{2} + 12 u \cdot 4 + 27 u}{64} d u$

Step 4.8

Subtract $16 u^{2}$ from $- 18 u^{2}$ .

$\int \frac{3 u^{3} - 34 u^{2} + 12 u \cdot 4 + 27 u}{64} d u$

Step 4.9

Multiply $4$ by $12$ .

$\int \frac{3 u^{3} - 34 u^{2} + 48 u + 27 u}{64} d u$

Step 4.10

Add $48 u$ and $27 u$ .

$\int \frac{3 u^{3} - 34 u^{2} + 75 u}{64} d u$

$\int \frac{3 u^{3} - 34 u^{2} + 75 u}{64} d u$

Step 5

Since $\frac{1}{64}$ is constant with respect to $u$ , move $\frac{1}{64}$ out of the integral.

$\frac{1}{64} \int 3 u^{3} - 34 u^{2} + 75 u d u$

Step 6

Split the single integral into multiple integrals.

$\frac{1}{64} (\int 3 u^{3} d u + \int - 34 u^{2} d u + \int 75 u d u)$

Step 7

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

$\frac{1}{64} (3 \int u^{3} d u + \int - 34 u^{2} d u + \int 75 u d u)$

Step 8

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

$\frac{1}{64} (3 (\frac{1}{4} u^{4} + C) + \int - 34 u^{2} d u + \int 75 u d u)$

Step 9

Since $- 34$ is constant with respect to $u$ , move $- 34$ out of the integral.

$\frac{1}{64} (3 (\frac{1}{4} u^{4} + C) - 34 \int u^{2} d u + \int 75 u d u)$

Step 10

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

$\frac{1}{64} (3 (\frac{1}{4} u^{4} + C) - 34 (\frac{1}{3} u^{3} + C) + \int 75 u d u)$

Step 11

Since $75$ is constant with respect to $u$ , move $75$ out of the integral.

$\frac{1}{64} (3 (\frac{1}{4} u^{4} + C) - 34 (\frac{1}{3} u^{3} + C) + 75 \int u d u)$

Step 12

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

$\frac{1}{64} (3 (\frac{1}{4} u^{4} + C) - 34 (\frac{1}{3} u^{3} + C) + 75 (\frac{1}{2} u^{2} + C))$

Step 13

Simplify.

$\frac{1}{64} (\frac{3 u^{4}}{4} - \frac{34 u^{3}}{3} + \frac{75 u^{2}}{2}) + C$

Step 14

Replace all occurrences of $u$ with $4 x + 3$ .

$\frac{1}{64} (\frac{3 {(4 x + 3)}^{4}}{4} - \frac{34 {(4 x + 3)}^{3}}{3} + \frac{75 {(4 x + 3)}^{2}}{2}) + C$

Step 15

Reorder terms.

$\frac{1}{64} (\frac{3}{4} {(4 x + 3)}^{4} - \frac{34}{3} {(4 x + 3)}^{3} + \frac{75}{2} {(4 x + 3)}^{2}) + C$

$[\begin{array}{c} x^{2} & \frac{1}{2} \\ \sqrt{π} & \int x d x \end{array}]$