Calculus Examples

$x (x + 9) \cdot (x + 6)$

Step 1

Let $u = x + 6$ . Then $d u = d x$ . Rewrite using $u$ and $d$ $u$ .

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Step 1.1

Let $u = x + 6$ . Find $\frac{d u}{d x}$ .

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Step 1.1.1

Differentiate $x + 6$ .

$\frac{d}{d x} [x + 6]$

Step 1.1.2

By the Sum Rule, the derivative of $x + 6$ with respect to $x$ is $\frac{d}{d x} [x] + \frac{d}{d x} [6]$ .

$\frac{d}{d x} [x] + \frac{d}{d x} [6]$

Step 1.1.3

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

$1 + \frac{d}{d x} [6]$

Step 1.1.4

Since $6$ is constant with respect to $x$ , the derivative of $6$ with respect to $x$ is $0$ .

$1 + 0$

Step 1.1.5

Add $1$ and $0$ .

$1$

Step 1.2

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

$\int (u - 6) (u - 6 + 9) \cdot u d u$

Step 2

Add $- 6$ and $9$ .

$\int (u - 6) (u + 3) u d u$

Step 3

Simplify.

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Step 3.1

Apply the distributive property.

$\int (u (u + 3) - 6 (u + 3)) u d u$

Step 3.2

Apply the distributive property.

$\int (u \cdot u + u \cdot 3 - 6 (u + 3)) u d u$

Step 3.3

Apply the distributive property.

$\int (u \cdot u + u \cdot 3 - 6 u - 6 \cdot 3) u d u$

Step 3.4

Apply the distributive property.

$\int (u \cdot u + u \cdot 3) u + (- 6 u - 6 \cdot 3) u d u$

Step 3.5

Apply the distributive property.

$\int u \cdot u \cdot u + u \cdot 3 u + (- 6 u - 6 \cdot 3) u d u$

Step 3.6

Apply the distributive property.

$\int u \cdot u \cdot u + u \cdot 3 u - 6 u \cdot u - 6 \cdot 3 u d u$

Step 3.7

Reorder $u$ and $3$ .

$\int u \cdot u \cdot u + 3 \cdot u \cdot u - 6 u \cdot u - 6 \cdot 3 u d u$

Step 3.8

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

$\int u^{1} u \cdot u + 3 u \cdot u - 6 u \cdot u - 6 \cdot 3 u d u$

Step 3.9

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

$\int u^{1} u^{1} u + 3 u \cdot u - 6 u \cdot u - 6 \cdot 3 u d u$

Step 3.10

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

$\int u^{1 + 1} u + 3 u \cdot u - 6 u \cdot u - 6 \cdot 3 u d u$

Step 3.11

Add $1$ and $1$ .

$\int u^{2} u + 3 u \cdot u - 6 u \cdot u - 6 \cdot 3 u d u$

Step 3.12

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

$\int u^{2} u^{1} + 3 u \cdot u - 6 u \cdot u - 6 \cdot 3 u d u$

Step 3.13

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

$\int u^{2 + 1} + 3 u \cdot u - 6 u \cdot u - 6 \cdot 3 u d u$

Step 3.14

Add $2$ and $1$ .

$\int u^{3} + 3 u \cdot u - 6 u \cdot u - 6 \cdot 3 u d u$

Step 3.15

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

$\int u^{3} + 3 (u^{1} u) - 6 u \cdot u - 6 \cdot 3 u d u$

Step 3.16

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

$\int u^{3} + 3 (u^{1} u^{1}) - 6 u \cdot u - 6 \cdot 3 u d u$

Step 3.17

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

$\int u^{3} + 3 u^{1 + 1} - 6 u \cdot u - 6 \cdot 3 u d u$

Step 3.18

Add $1$ and $1$ .

$\int u^{3} + 3 u^{2} - 6 u \cdot u - 6 \cdot 3 u d u$

Step 3.19

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

$\int u^{3} + 3 u^{2} - 6 (u^{1} u) - 6 \cdot 3 u d u$

Step 3.20

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

$\int u^{3} + 3 u^{2} - 6 (u^{1} u^{1}) - 6 \cdot 3 u d u$

Step 3.21

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

$\int u^{3} + 3 u^{2} - 6 u^{1 + 1} - 6 \cdot 3 u d u$

Step 3.22

Add $1$ and $1$ .

$\int u^{3} + 3 u^{2} - 6 u^{2} - 6 \cdot 3 u d u$

Step 3.23

Multiply $- 6$ by $3$ .

$\int u^{3} + 3 u^{2} - 6 u^{2} - 18 u d u$

Step 3.24

Subtract $6 u^{2}$ from $3 u^{2}$ .

$\int u^{3} - 3 u^{2} - 18 u d u$

Step 4

Split the single integral into multiple integrals.

$\int u^{3} d u + \int - 3 u^{2} d u + \int - 18 u d u$

Step 5

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

$\frac{1}{4} u^{4} + C + \int - 3 u^{2} d u + \int - 18 u d u$

Step 6

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

$\frac{1}{4} u^{4} + C - 3 \int u^{2} d u + \int - 18 u d u$

Step 7

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

$\frac{1}{4} u^{4} + C - 3 (\frac{1}{3} u^{3} + C) + \int - 18 u d u$

Step 8

Since $- 18$ is constant with respect to $u$ , move $- 18$ out of the integral.

$\frac{1}{4} u^{4} + C - 3 (\frac{1}{3} u^{3} + C) - 18 \int u d u$

Step 9

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

$\frac{1}{4} u^{4} + C - 3 (\frac{1}{3} u^{3} + C) - 18 (\frac{1}{2} u^{2} + C)$

Step 10

Simplify.

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Step 10.1

Simplify.

$\frac{u^{4}}{4} - u^{3} - 18 (\frac{1}{2} u^{2}) + C$

Step 10.2

Simplify.

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Step 10.2.1

Combine $\frac{1}{2}$ and $u^{2}$ .

$\frac{u^{4}}{4} - u^{3} - 18 \frac{u^{2}}{2} + C$

Step 10.2.2

Combine $- 18$ and $\frac{u^{2}}{2}$ .

$\frac{u^{4}}{4} - u^{3} + \frac{- 18 u^{2}}{2} + C$

Step 10.2.3

Cancel the common factor of $- 18$ and $2$ .

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Step 10.2.3.1

Factor $2$ out of $- 18 u^{2}$ .

$\frac{u^{4}}{4} - u^{3} + \frac{2 (- 9 u^{2})}{2} + C$

Step 10.2.3.2

Cancel the common factors.

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Step 10.2.3.2.1

Factor $2$ out of $2$ .

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

Step 10.2.3.2.2

Cancel the common factor.

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

Step 10.2.3.2.3

Rewrite the expression.

$\frac{u^{4}}{4} - u^{3} + \frac{- 9 u^{2}}{1} + C$

Step 10.2.3.2.4

Divide $- 9 u^{2}$ by $1$ .

$\frac{u^{4}}{4} - u^{3} - 9 u^{2} + C$

Step 11

Replace all occurrences of $u$ with $x + 6$ .

$\frac{{(x + 6)}^{4}}{4} - {(x + 6)}^{3} - 9 {(x + 6)}^{2} + C$

Step 12

Reorder terms.

$\frac{1}{4} {(x + 6)}^{4} - {(x + 6)}^{3} - 9 {(x + 6)}^{2} + C$