Calculus Examples

$\int 4 x (2 x - 3) (3 x + 1) d x$

Step 1

Since $4$ is constant with respect to $x$ , move $4$ out of the integral.

$4 \int (x (2 x - 3)) (3 x + 1) d x$

Step 2

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

Tap for more steps...

Step 2.1

Let $u = 3 x + 1$ . Find $\frac{d u}{d x}$ .

Tap for more steps...

Step 2.1.1

Differentiate $3 x + 1$ .

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

Step 2.1.2

By the Sum Rule, the derivative of $3 x + 1$ with respect to $x$ is $\frac{d}{d x} [3 x] + \frac{d}{d x} [1]$ .

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

Step 2.1.3

Evaluate $\frac{d}{d x} [3 x]$ .

Tap for more steps...

Step 2.1.3.1

Since $3$ is constant with respect to $x$ , the derivative of $3 x$ with respect to $x$ is $3 \frac{d}{d x} [x]$ .

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

Step 2.1.3.2

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

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

Step 2.1.3.3

Multiply $3$ by $1$ .

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

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

Step 2.1.4

Differentiate using the Constant Rule.

Tap for more steps...

Step 2.1.4.1

Since $1$ is constant with respect to $x$ , the derivative of $1$ with respect to $x$ is $0$ .

$3 + 0$

Step 2.1.4.2

Add $3$ and $0$ .

$3$

$3$

$3$

Step 2.2

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

$4 \int (\frac{u}{3} - \frac{1}{3}) (2 (\frac{u}{3} - \frac{1}{3}) - 3) u \frac{1}{3} d u$

$4 \int (\frac{u}{3} - \frac{1}{3}) (2 (\frac{u}{3} - \frac{1}{3}) - 3) u \frac{1}{3} d u$

Step 3

Combine $\frac{1}{3}$ and $u$ .

$4 \int (\frac{u}{3} - \frac{1}{3}) (2 (\frac{u}{3} - \frac{1}{3}) - 3) \frac{u}{3} d u$

Step 4

Simplify.

Tap for more steps...

Step 4.1

Apply the distributive property.

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

Step 4.2

Apply the distributive property.

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

Step 4.3

Apply the distributive property.

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

Step 4.4

Apply the distributive property.

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

Step 4.5

Apply the distributive property.

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

Step 4.6

Apply the distributive property.

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

Step 4.7

Apply the distributive property.

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

Step 4.8

Apply the distributive property.

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

Step 4.9

Apply the distributive property.

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

Step 4.10

Apply the distributive property.

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

Step 4.11

Apply the distributive property.

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

Step 4.12

Reorder $\frac{u}{3}$ and $2$ .

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

Step 4.13

Move parentheses.

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

Step 4.14

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

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

Step 4.15

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

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

Step 4.16

Move $\frac{1}{3}$ .

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

Step 4.17

Move parentheses.

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

Step 4.18

Move $\frac{1}{3}$ .

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

Step 4.19

Move $\frac{1}{3}$ .

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

Step 4.20

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

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

Step 4.21

Multiply $\frac{2 u}{3}$ by $\frac{u}{3}$ .

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

Step 4.22

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

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

Step 4.23

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

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

Step 4.24

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

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

Step 4.25

Add $1$ and $1$ .

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

Step 4.26

Multiply $3$ by $3$ .

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

Step 4.27

Multiply $\frac{2 u^{2}}{9}$ by $\frac{u}{3}$ .

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

Step 4.28

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

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

Step 4.29

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

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

Step 4.30

Add $2$ and $1$ .

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

Step 4.31

Multiply $9$ by $3$ .

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

Step 4.32

Multiply $2$ by $- 1$ .

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

Step 4.33

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

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

Step 4.34

Multiply $\frac{- 2 u}{3}$ by $\frac{1}{3}$ .

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

Step 4.35

Multiply $3$ by $3$ .

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

Step 4.36

Multiply $\frac{- 2 u}{9}$ by $\frac{u}{3}$ .

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

Step 4.37

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

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

Step 4.38

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

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

Step 4.39

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

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

Step 4.40

Add $1$ and $1$ .

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

Step 4.41

Multiply $9$ by $3$ .

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

Step 4.42

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

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

Step 4.43

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

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

Step 4.44

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

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

Step 4.45

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

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

Step 4.46

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

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

Step 4.47

Add $1$ and $1$ .

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

Step 4.48

Multiply $3$ by $3$ .

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

Step 4.49

To write $\frac{- 3 u^{2}}{9}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

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

Step 4.50

Write each expression with a common denominator of $27$ , by multiplying each by an appropriate factor of $1$ .

Tap for more steps...

Step 4.50.1

Multiply $\frac{- 3 u^{2}}{9}$ by $\frac{3}{3}$ .

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

Step 4.50.2

Multiply $9$ by $3$ .

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

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

Step 4.51

Combine the numerators over the common denominator.

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

Step 4.52

Combine the numerators over the common denominator.

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

Step 4.53

Multiply $- 1$ by $2$ .

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

Step 4.54

Combine $- 2$ and $\frac{1}{3}$ .

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

Step 4.55

Multiply $\frac{- 2}{3}$ by $\frac{u}{3}$ .

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

Step 4.56

Multiply $3$ by $3$ .

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

Step 4.57

Multiply $\frac{- 2 u}{9}$ by $\frac{u}{3}$ .

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

Step 4.58

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

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

Step 4.59

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

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

Step 4.60

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

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

Step 4.61

Add $1$ and $1$ .

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

Step 4.62

Multiply $9$ by $3$ .

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

Step 4.63

Multiply $- 1$ by $2$ .

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

Step 4.64

Multiply $- 2$ by $- 1$ .

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

Step 4.65

Combine $2$ and $\frac{1}{3}$ .

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

Step 4.66

Multiply $\frac{2}{3}$ by $\frac{1}{3}$ .

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

Step 4.67

Multiply $3$ by $3$ .

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

Step 4.68

Multiply $\frac{2}{9}$ by $\frac{u}{3}$ .

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

Step 4.69

Multiply $9$ by $3$ .

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

Step 4.70

Multiply $- 1$ by $- 3$ .

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

Step 4.71

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

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

Step 4.72

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

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

Step 4.73

Multiply $3$ by $3$ .

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

Step 4.74

To write $\frac{3 u}{9}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

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

Step 4.75

Write each expression with a common denominator of $27$ , by multiplying each by an appropriate factor of $1$ .

Tap for more steps...

Step 4.75.1

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

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

Step 4.75.2

Multiply $9$ by $3$ .

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

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

Step 4.76

Combine the numerators over the common denominator.

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

Step 4.77

Combine the numerators over the common denominator.

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

Step 4.78

Combine the numerators over the common denominator.

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

Step 4.79

Reorder $2 u$ and $3 u \cdot 3$ .

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

Step 4.80

Move $- 3 u^{2} \cdot 3$ .

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

Step 4.81

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

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

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

Step 5

Simplify.

Tap for more steps...

Step 5.1

Multiply $3$ by $- 3$ .

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

Step 5.2

Subtract $9 u^{2}$ from $- 4 u^{2}$ .

$4 \int \frac{2 u^{3} - 13 u^{2} + 3 u \cdot 3 + 2 u}{27} d u$

Step 5.3

Multiply $3$ by $3$ .

$4 \int \frac{2 u^{3} - 13 u^{2} + 9 u + 2 u}{27} d u$

Step 5.4

Add $9 u$ and $2 u$ .

$4 \int \frac{2 u^{3} - 13 u^{2} + 11 u}{27} d u$

$4 \int \frac{2 u^{3} - 13 u^{2} + 11 u}{27} d u$

Step 6

Since $\frac{1}{27}$ is constant with respect to $u$ , move $\frac{1}{27}$ out of the integral.

$4 (\frac{1}{27} \int 2 u^{3} - 13 u^{2} + 11 u d u)$

Step 7

Combine $\frac{1}{27}$ and $4$ .

$\frac{4}{27} \int 2 u^{3} - 13 u^{2} + 11 u d u$

Step 8

Split the single integral into multiple integrals.

$\frac{4}{27} (\int 2 u^{3} d u + \int - 13 u^{2} d u + \int 11 u d u)$

Step 9

Since $2$ is constant with respect to $u$ , move $2$ out of the integral.

$\frac{4}{27} (2 \int u^{3} d u + \int - 13 u^{2} d u + \int 11 u d u)$

Step 10

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

$\frac{4}{27} (2 (\frac{1}{4} u^{4} + C) + \int - 13 u^{2} d u + \int 11 u d u)$

Step 11

Since $- 13$ is constant with respect to $u$ , move $- 13$ out of the integral.

$\frac{4}{27} (2 (\frac{1}{4} u^{4} + C) - 13 \int u^{2} d u + \int 11 u d u)$

Step 12

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

$\frac{4}{27} (2 (\frac{1}{4} u^{4} + C) - 13 (\frac{1}{3} u^{3} + C) + \int 11 u d u)$

Step 13

Since $11$ is constant with respect to $u$ , move $11$ out of the integral.

$\frac{4}{27} (2 (\frac{1}{4} u^{4} + C) - 13 (\frac{1}{3} u^{3} + C) + 11 \int u d u)$

Step 14

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

$\frac{4}{27} (2 (\frac{1}{4} u^{4} + C) - 13 (\frac{1}{3} u^{3} + C) + 11 (\frac{1}{2} u^{2} + C))$

Step 15

Simplify.

$\frac{4}{27} (\frac{u^{4}}{2} - \frac{13 u^{3}}{3} + \frac{11 u^{2}}{2}) + C$

Step 16

Replace all occurrences of $u$ with $3 x + 1$ .

$\frac{4}{27} (\frac{{(3 x + 1)}^{4}}{2} - \frac{13 {(3 x + 1)}^{3}}{3} + \frac{11 {(3 x + 1)}^{2}}{2}) + C$

Step 17

Reorder terms.

$\frac{4}{27} (\frac{1}{2} {(3 x + 1)}^{4} - \frac{13}{3} {(3 x + 1)}^{3} + \frac{11}{2} {(3 x + 1)}^{2}) + C$

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