Calculus Examples

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

Step 1

Let $u = 4 x - 1$ . 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 $u = 4 x - 1$ . Find $\frac{d u}{d x}$ .

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Step 1.1.1

Differentiate $4 x - 1$ .

$\frac{d}{d x} [4 x - 1]$

Step 1.1.2

By the Sum Rule, the derivative of $4 x - 1$ with respect to $x$ is $\frac{d}{d x} [4 x] + \frac{d}{d x} [- 1]$ .

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

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

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

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 $n = 1$ .

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

Step 1.1.3.3

Multiply $4$ by $1$ .

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

Step 1.1.4

Differentiate using the Constant Rule.

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Step 1.1.4.1

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

$4 + 0$

Step 1.1.4.2

Add $4$ and $0$ .

$4$

Step 1.2

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

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

Step 2

Simplify.

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Step 2.1

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

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

Step 2.2

Combine $2$ and $\frac{4}{4}$ .

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

Step 2.3

Combine the numerators over the common denominator.

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

Step 2.4

Simplify the numerator.

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Step 2.4.1

Multiply $2$ by $4$ .

$\int (\frac{u}{4} + \frac{1}{4}) (\frac{u}{4} + \frac{1 + 8}{4}) u \frac{1}{4} d u$

Step 2.4.2

Add $1$ and $8$ .

$\int (\frac{u}{4} + \frac{1}{4}) (\frac{u}{4} + \frac{9}{4}) u \frac{1}{4} d u$

Step 2.5

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

$\int (\frac{u}{4} + \frac{1}{4}) (\frac{u}{4} + \frac{9}{4}) \frac{u}{4} d u$

Step 3

Simplify.

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Step 3.1

Apply the distributive property.

$\int (\frac{u}{4} (\frac{u}{4} + \frac{9}{4}) + \frac{1}{4} (\frac{u}{4} + \frac{9}{4})) \frac{u}{4} d u$

Step 3.2

Apply the distributive property.

$\int (\frac{u}{4} \cdot \frac{u}{4} + \frac{u}{4} \cdot \frac{9}{4} + \frac{1}{4} (\frac{u}{4} + \frac{9}{4})) \frac{u}{4} d u$

Step 3.3

Apply the distributive property.

$\int (\frac{u}{4} \cdot \frac{u}{4} + \frac{u}{4} \cdot \frac{9}{4} + \frac{1}{4} \cdot \frac{u}{4} + \frac{1}{4} \cdot \frac{9}{4}) \frac{u}{4} d u$

Step 3.4

Apply the distributive property.

$\int (\frac{u}{4} \cdot \frac{u}{4} + \frac{u}{4} \cdot \frac{9}{4}) \frac{u}{4} + (\frac{1}{4} \cdot \frac{u}{4} + \frac{1}{4} \cdot \frac{9}{4}) \frac{u}{4} d u$

Step 3.5

Apply the distributive property.

$\int \frac{u}{4} \cdot \frac{u}{4} \frac{u}{4} + \frac{u}{4} \cdot \frac{9}{4} \frac{u}{4} + (\frac{1}{4} \cdot \frac{u}{4} + \frac{1}{4} \cdot \frac{9}{4}) \frac{u}{4} d u$

Step 3.6

Apply the distributive property.

$\int \frac{u}{4} \cdot \frac{u}{4} \frac{u}{4} + \frac{u}{4} \cdot \frac{9}{4} \frac{u}{4} + \frac{1}{4} \cdot \frac{u}{4} \frac{u}{4} + \frac{1}{4} \cdot \frac{9}{4} \frac{u}{4} d u$

Step 3.7

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

$\int \frac{u \cdot u}{4 \cdot 4} \cdot \frac{u}{4} + \frac{u}{4} \cdot \frac{9}{4} \frac{u}{4} + \frac{1}{4} \cdot \frac{u}{4} \frac{u}{4} + \frac{1}{4} \cdot \frac{9}{4} \frac{u}{4} d u$

Step 3.8

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

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

Step 3.9

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

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

Step 3.10

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

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

Step 3.11

Add $1$ and $1$ .

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

Step 3.12

Multiply $4$ by $4$ .

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

Step 3.13

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

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

Step 3.14

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

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

Step 3.15

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

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

Step 3.16

Add $2$ and $1$ .

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

Step 3.17

Multiply $16$ by $4$ .

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

Step 3.18

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

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

Step 3.19

Multiply $4$ by $4$ .

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

Step 3.20

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

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

Step 3.21

Multiply $16$ by $4$ .

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

Step 3.22

Multiply $\frac{1}{4}$ by $\frac{u}{4}$ .

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

Step 3.23

Multiply $4$ by $4$ .

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

Step 3.24

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

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

Step 3.25

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

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

Step 3.26

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

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

Step 3.27

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

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

Step 3.28

Add $1$ and $1$ .

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

Step 3.29

Multiply $16$ by $4$ .

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

Step 3.30

Multiply $\frac{1}{4}$ by $\frac{9}{4}$ .

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

Step 3.31

Multiply $4$ by $4$ .

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

Step 3.32

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

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

Step 3.33

Multiply $16$ by $4$ .

$\int \frac{u^{3}}{64} + \frac{u \cdot 9 u}{64} + \frac{u^{2}}{64} + \frac{9 u}{64} d u$

Step 4

Simplify.

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Step 4.1

Move $9$ to the left of $u$ .

$\int \frac{u^{3}}{64} + \frac{9 \cdot u \cdot u}{64} + \frac{u^{2}}{64} + \frac{9 u}{64} d u$

Step 4.2

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

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

Step 4.3

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

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

Step 4.4

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

$\int \frac{u^{3}}{64} + \frac{9 u^{1 + 1}}{64} + \frac{u^{2}}{64} + \frac{9 u}{64} d u$

Step 4.5

Add $1$ and $1$ .

$\int \frac{u^{3}}{64} + \frac{9 u^{2}}{64} + \frac{u^{2}}{64} + \frac{9 u}{64} d u$

Step 5

Split the single integral into multiple integrals.

$\int \frac{u^{3}}{64} d u + \int \frac{9 u^{2}}{64} d u + \int \frac{u^{2}}{64} d u + \int \frac{9 u}{64} d u$

Step 6

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

$\frac{1}{64} \int u^{3} d u + \int \frac{9 u^{2}}{64} d u + \int \frac{u^{2}}{64} d u + \int \frac{9 u}{64} d u$

Step 7

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

$\frac{1}{64} (\frac{1}{4} u^{4} + C) + \int \frac{9 u^{2}}{64} d u + \int \frac{u^{2}}{64} d u + \int \frac{9 u}{64} d u$

Step 8

Since $\frac{9}{64}$ is constant with respect to $u$ , move $\frac{9}{64}$ out of the integral.

$\frac{1}{64} (\frac{1}{4} u^{4} + C) + \frac{9}{64} \int u^{2} d u + \int \frac{u^{2}}{64} d u + \int \frac{9 u}{64} d u$

Step 9

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

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

Step 10

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

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

Step 11

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

$\frac{1}{64} (\frac{1}{4} u^{4} + C) + \frac{9}{64} (\frac{1}{3} u^{3} + C) + \frac{1}{64} (\frac{1}{3} u^{3} + C) + \int \frac{9 u}{64} d u$

Step 12

Since $\frac{9}{64}$ is constant with respect to $u$ , move $\frac{9}{64}$ out of the integral.

$\frac{1}{64} (\frac{1}{4} u^{4} + C) + \frac{9}{64} (\frac{1}{3} u^{3} + C) + \frac{1}{64} (\frac{1}{3} u^{3} + C) + \frac{9}{64} \int u d u$

Step 13

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

$\frac{1}{64} (\frac{1}{4} u^{4} + C) + \frac{9}{64} (\frac{1}{3} u^{3} + C) + \frac{1}{64} (\frac{1}{3} u^{3} + C) + \frac{9}{64} (\frac{1}{2} u^{2} + C)$

Step 14

Simplify.

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Step 14.1

Simplify.

$\frac{u^{4}}{256} + \frac{3 u^{3}}{64} + \frac{u^{3}}{192} + \frac{9}{64} (\frac{1}{2} u^{2}) + C$

Step 14.2

Simplify.

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Step 14.2.1

To write $\frac{3 u^{3}}{64}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$\frac{u^{4}}{256} + \frac{3 u^{3}}{64} \cdot \frac{3}{3} + \frac{u^{3}}{192} + \frac{9}{64} (\frac{1}{2} u^{2}) + C$

Step 14.2.2

Write each expression with a common denominator of $192$ , by multiplying each by an appropriate factor of $1$ .

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Step 14.2.2.1

Multiply $\frac{3 u^{3}}{64}$ by $\frac{3}{3}$ .

$\frac{u^{4}}{256} + \frac{3 u^{3} \cdot 3}{64 \cdot 3} + \frac{u^{3}}{192} + \frac{9}{64} (\frac{1}{2} u^{2}) + C$

Step 14.2.2.2

Multiply $64$ by $3$ .

$\frac{u^{4}}{256} + \frac{3 u^{3} \cdot 3}{192} + \frac{u^{3}}{192} + \frac{9}{64} (\frac{1}{2} u^{2}) + C$

Step 14.2.3

Combine the numerators over the common denominator.

$\frac{u^{4}}{256} + \frac{3 u^{3} \cdot 3 + u^{3}}{192} + \frac{9}{64} (\frac{1}{2} u^{2}) + C$

Step 14.2.4

Multiply $3$ by $3$ .

$\frac{u^{4}}{256} + \frac{9 u^{3} + u^{3}}{192} + \frac{9}{64} (\frac{1}{2} u^{2}) + C$

Step 14.2.5

Add $9 u^{3}$ and $u^{3}$ .

$\frac{u^{4}}{256} + \frac{10 u^{3}}{192} + \frac{9}{64} (\frac{1}{2} u^{2}) + C$

Step 14.2.6

Cancel the common factor of $10$ and $192$ .

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Step 14.2.6.1

Factor $2$ out of $10 u^{3}$ .

$\frac{u^{4}}{256} + \frac{2 (5 u^{3})}{192} + \frac{9}{64} (\frac{1}{2} u^{2}) + C$

Step 14.2.6.2

Cancel the common factors.

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Step 14.2.6.2.1

Factor $2$ out of $192$ .

$\frac{u^{4}}{256} + \frac{2 (5 u^{3})}{2 (96)} + \frac{9}{64} (\frac{1}{2} u^{2}) + C$

Step 14.2.6.2.2

Cancel the common factor.

$\frac{u^{4}}{256} + \frac{2 (5 u^{3})}{2 \cdot 96} + \frac{9}{64} (\frac{1}{2} u^{2}) + C$

Step 14.2.6.2.3

Rewrite the expression.

$\frac{u^{4}}{256} + \frac{5 u^{3}}{96} + \frac{9}{64} (\frac{1}{2} u^{2}) + C$

Step 14.2.7

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

$\frac{u^{4}}{256} + \frac{5 u^{3}}{96} + \frac{9}{64} \cdot \frac{u^{2}}{2} + C$

Step 14.2.8

Multiply $\frac{9}{64}$ by $\frac{u^{2}}{2}$ .

$\frac{u^{4}}{256} + \frac{5 u^{3}}{96} + \frac{9 u^{2}}{64 \cdot 2} + C$

Step 14.2.9

Multiply $64$ by $2$ .

$\frac{u^{4}}{256} + \frac{5 u^{3}}{96} + \frac{9 u^{2}}{128} + C$

Step 15

Replace all occurrences of $u$ with $4 x - 1$ .

$\frac{{(4 x - 1)}^{4}}{256} + \frac{5 {(4 x - 1)}^{3}}{96} + \frac{9 {(4 x - 1)}^{2}}{128} + C$

Step 16

Reorder terms.

$\frac{1}{256} {(4 x - 1)}^{4} + \frac{5}{96} {(4 x - 1)}^{3} + \frac{9}{128} {(4 x - 1)}^{2} + C$