Calculus Examples

$\int_{0}^{2} {(x - 6)}^{2} d x$

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

Substitute the lower limit in for $x$ in $u = x - 6$ .

$u_{lower} = 0 - 6$

Step 1.3

Subtract $6$ from $0$ .

$u_{lower} = - 6$

Step 1.4

Substitute the upper limit in for $x$ in $u = x - 6$ .

$u_{upper} = 2 - 6$

Step 1.5

Subtract $6$ from $2$ .

$u_{upper} = - 4$

Step 1.6

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

$u_{lower} = - 6$

$u_{upper} = - 4$

Step 1.7

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

$\int_{- 6}^{- 4} u^{2} d u$

Step 2

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

$\frac{1}{3} u^{3}]_{- 6}^{- 4}$

Step 3

Substitute and simplify.

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Step 3.1

Evaluate $\frac{1}{3} u^{3}$ at $- 4$ and at $- 6$ .

$(\frac{1}{3} {(- 4)}^{3}) - \frac{1}{3} {(- 6)}^{3}$

Step 3.2

Simplify.

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Step 3.2.1

Raise $- 4$ to the power of $3$ .

$\frac{1}{3} \cdot - 64 - \frac{1}{3} {(- 6)}^{3}$

Step 3.2.2

Combine $\frac{1}{3}$ and $- 64$ .

$\frac{- 64}{3} - \frac{1}{3} {(- 6)}^{3}$

Step 3.2.3

Move the negative in front of the fraction.

$- \frac{64}{3} - \frac{1}{3} {(- 6)}^{3}$

Step 3.2.4

Raise $- 6$ to the power of $3$ .

$- \frac{64}{3} - \frac{1}{3} \cdot - 216$

Step 3.2.5

Multiply $- 216$ by $- 1$ .

$- \frac{64}{3} + 216 (\frac{1}{3})$

Step 3.2.6

Combine $216$ and $\frac{1}{3}$ .

$- \frac{64}{3} + \frac{216}{3}$

Step 3.2.7

Cancel the common factor of $216$ and $3$ .

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Step 3.2.7.1

Factor $3$ out of $216$ .

$- \frac{64}{3} + \frac{3 \cdot 72}{3}$

Step 3.2.7.2

Cancel the common factors.

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Step 3.2.7.2.1

Factor $3$ out of $3$ .

$- \frac{64}{3} + \frac{3 \cdot 72}{3 (1)}$

Step 3.2.7.2.2

Cancel the common factor.

$- \frac{64}{3} + \frac{3 \cdot 72}{3 \cdot 1}$

Step 3.2.7.2.3

Rewrite the expression.

$- \frac{64}{3} + \frac{72}{1}$

Step 3.2.7.2.4

Divide $72$ by $1$ .

$- \frac{64}{3} + 72$

Step 3.2.8

To write $72$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$- \frac{64}{3} + 72 \cdot \frac{3}{3}$

Step 3.2.9

Combine $72$ and $\frac{3}{3}$ .

$- \frac{64}{3} + \frac{72 \cdot 3}{3}$

Step 3.2.10

Combine the numerators over the common denominator.

$\frac{- 64 + 72 \cdot 3}{3}$

Step 3.2.11

Simplify the numerator.

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Step 3.2.11.1

Multiply $72$ by $3$ .

$\frac{- 64 + 216}{3}$

Step 3.2.11.2

Add $- 64$ and $216$ .

$\frac{152}{3}$

Step 4

The result can be shown in multiple forms.

Exact Form:

$\frac{152}{3}$

Decimal Form:

$50.\overline{6}$

Mixed Number Form:

$50 \frac{2}{3}$

Step 5