Calculus Examples

$\int_{1}^{- 1} {(12 x + 11)}^{2} d x$

Step 1

Let $u = 12 x + 11$ . Then $d u = 12 d x$ , so $\frac{1}{12} d u = d x$ . Rewrite using $u$ and $d$ $u$ .

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Step 1.1

Let $u = 12 x + 11$ . Find $\frac{d u}{d x}$ .

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Step 1.1.1

Differentiate $12 x + 11$ .

$\frac{d}{d x} [12 x + 11]$

Step 1.1.2

By the Sum Rule, the derivative of $12 x + 11$ with respect to $x$ is $\frac{d}{d x} [12 x] + \frac{d}{d x} [11]$ .

$\frac{d}{d x} [12 x] + \frac{d}{d x} [11]$

Step 1.1.3

Evaluate $\frac{d}{d x} [12 x]$ .

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Step 1.1.3.1

Since $12$ is constant with respect to $x$ , the derivative of $12 x$ with respect to $x$ is $12 \frac{d}{d x} [x]$ .

$12 \frac{d}{d x} [x] + \frac{d}{d x} [11]$

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 $n = 1$ .

$12 \cdot 1 + \frac{d}{d x} [11]$

Step 1.1.3.3

Multiply $12$ by $1$ .

$12 + \frac{d}{d x} [11]$

Step 1.1.4

Differentiate using the Constant Rule.

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Step 1.1.4.1

Since $11$ is constant with respect to $x$ , the derivative of $11$ with respect to $x$ is $0$ .

$12 + 0$

Step 1.1.4.2

Add $12$ and $0$ .

$12$

Step 1.2

Substitute the lower limit in for $x$ in $u = 12 x + 11$ .

$u_{lower} = 12 \cdot 1 + 11$

Step 1.3

Simplify.

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Step 1.3.1

Multiply $12$ by $1$ .

$u_{lower} = 12 + 11$

Step 1.3.2

Add $12$ and $11$ .

$u_{lower} = 23$

Step 1.4

Substitute the upper limit in for $x$ in $u = 12 x + 11$ .

$u_{upper} = 12 \cdot - 1 + 11$

Step 1.5

Simplify.

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Step 1.5.1

Multiply $12$ by $- 1$ .

$u_{upper} = - 12 + 11$

Step 1.5.2

Add $- 12$ and $11$ .

$u_{upper} = - 1$

Step 1.6

The values found for $u_{lower}$ and $u_{upper}$ will be used to evaluate the definite integral.

$u_{lower} = 23$

$u_{upper} = - 1$

Step 1.7

Rewrite the problem using $u$ , $d u$ , and the new limits of integration.

$\int_{23}^{- 1} u^{2} \frac{1}{12} d u$

Step 2

Combine $u^{2}$ and $\frac{1}{12}$ .

$\int_{23}^{- 1} \frac{u^{2}}{12} d u$

Step 3

Since $\frac{1}{12}$ is constant with respect to $u$ , move $\frac{1}{12}$ out of the integral.

$\frac{1}{12} \int_{23}^{- 1} u^{2} d u$

Step 4

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

$\frac{1}{12} \frac{1}{3} u^{3}]_{23}^{- 1}$

Step 5

Substitute and simplify.

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Step 5.1

Evaluate $\frac{1}{3} u^{3}$ at $- 1$ and at $23$ .

$\frac{1}{12} ((\frac{1}{3} {(- 1)}^{3}) - \frac{1}{3} \cdot 23^{3})$

Step 5.2

Simplify.

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Step 5.2.1

Raise $- 1$ to the power of $3$ .

$\frac{1}{12} (\frac{1}{3} \cdot - 1 - \frac{1}{3} \cdot 23^{3})$

Step 5.2.2

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

$\frac{1}{12} (\frac{- 1}{3} - \frac{1}{3} \cdot 23^{3})$

Step 5.2.3

Move the negative in front of the fraction.

$\frac{1}{12} (- \frac{1}{3} - \frac{1}{3} \cdot 23^{3})$

Step 5.2.4

Raise $23$ to the power of $3$ .

$\frac{1}{12} (- \frac{1}{3} - \frac{1}{3} \cdot 12167)$

Step 5.2.5

Multiply $12167$ by $- 1$ .

$\frac{1}{12} (- \frac{1}{3} - 12167 (\frac{1}{3}))$

Step 5.2.6

Combine $- 12167$ and $\frac{1}{3}$ .

$\frac{1}{12} (- \frac{1}{3} + \frac{- 12167}{3})$

Step 5.2.7

Move the negative in front of the fraction.

$\frac{1}{12} (- \frac{1}{3} - \frac{12167}{3})$

Step 5.2.8

Combine the numerators over the common denominator.

$\frac{1}{12} \cdot \frac{- 1 - 12167}{3}$

Step 5.2.9

Subtract $12167$ from $- 1$ .

$\frac{1}{12} \cdot \frac{- 12168}{3}$

Step 5.2.10

Cancel the common factor of $- 12168$ and $3$ .

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Step 5.2.10.1

Factor $3$ out of $- 12168$ .

$\frac{1}{12} \cdot \frac{3 \cdot - 4056}{3}$

Step 5.2.10.2

Cancel the common factors.

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Step 5.2.10.2.1

Factor $3$ out of $3$ .

$\frac{1}{12} \cdot \frac{3 \cdot - 4056}{3 (1)}$

Step 5.2.10.2.2

Cancel the common factor.

$\frac{1}{12} \cdot \frac{3 \cdot - 4056}{3 \cdot 1}$

Step 5.2.10.2.3

Rewrite the expression.

$\frac{1}{12} \cdot \frac{- 4056}{1}$

Step 5.2.10.2.4

Divide $- 4056$ by $1$ .

$\frac{1}{12} \cdot - 4056$

Step 5.2.11

Combine $\frac{1}{12}$ and $- 4056$ .

$\frac{- 4056}{12}$

Step 5.2.12

Cancel the common factor of $- 4056$ and $12$ .

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Step 5.2.12.1

Factor $12$ out of $- 4056$ .

$\frac{12 \cdot - 338}{12}$

Step 5.2.12.2

Cancel the common factors.

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Step 5.2.12.2.1

Factor $12$ out of $12$ .

$\frac{12 \cdot - 338}{12 (1)}$

Step 5.2.12.2.2

Cancel the common factor.

$\frac{12 \cdot - 338}{12 \cdot 1}$

Step 5.2.12.2.3

Rewrite the expression.

$\frac{- 338}{1}$

Step 5.2.12.2.4

Divide $- 338$ by $1$ .

$- 338$

Step 6