Calculus Examples

$\int_{0}^{2} {(1 - 4 x)}^{2} d x$

Step 1

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

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Step 1.1.1

Differentiate $1 - 4 x$ .

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

Step 1.1.2

Differentiate.

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Step 1.1.2.1

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

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

Step 1.1.2.2

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

$0 + \frac{d}{d x} [- 4 x]$

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

$0 - 4 \frac{d}{d x} [x]$

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 $n = 1$ .

$0 - 4 \cdot 1$

Step 1.1.3.3

Multiply $- 4$ by $1$ .

$0 - 4$

Step 1.1.4

Subtract $4$ from $0$ .

$- 4$

Step 1.2

Substitute the lower limit in for $x$ in $u = 1 - 4 x$ .

$u_{lower} = 1 - 4 \cdot 0$

Step 1.3

Simplify.

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Step 1.3.1

Multiply $- 4$ by $0$ .

$u_{lower} = 1 + 0$

Step 1.3.2

Add $1$ and $0$ .

$u_{lower} = 1$

Step 1.4

Substitute the upper limit in for $x$ in $u = 1 - 4 x$ .

$u_{upper} = 1 - 4 \cdot 2$

Step 1.5

Simplify.

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Step 1.5.1

Multiply $- 4$ by $2$ .

$u_{upper} = 1 - 8$

Step 1.5.2

Subtract $8$ from $1$ .

$u_{upper} = - 7$

Step 1.6

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

$u_{lower} = 1$

$u_{upper} = - 7$

Step 1.7

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

$\int_{1}^{- 7} u^{2} \frac{1}{- 4} d u$

Step 2

Simplify.

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Step 2.1

Move the negative in front of the fraction.

$\int_{1}^{- 7} u^{2} (- \frac{1}{4}) d u$

Step 2.2

Combine $u^{2}$ and $\frac{1}{4}$ .

$\int_{1}^{- 7} - \frac{u^{2}}{4} d u$

Step 3

Since $- 1$ is constant with respect to $u$ , move $- 1$ out of the integral.

$- \int_{1}^{- 7} \frac{u^{2}}{4} d u$

Step 4

Since $\frac{1}{4}$ is constant with respect to $u$ , move $\frac{1}{4}$ out of the integral.

$- (\frac{1}{4} \int_{1}^{- 7} u^{2} d u)$

Step 5

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

$- \frac{1}{4} \frac{1}{3} u^{3}]_{1}^{- 7}$

Step 6

Substitute and simplify.

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Step 6.1

Evaluate $\frac{1}{3} u^{3}$ at $- 7$ and at $1$ .

$- \frac{1}{4} ((\frac{1}{3} {(- 7)}^{3}) - \frac{1}{3} \cdot 1^{3})$

Step 6.2

Simplify.

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Step 6.2.1

Raise $- 7$ to the power of $3$ .

$- \frac{1}{4} (\frac{1}{3} \cdot - 343 - \frac{1}{3} \cdot 1^{3})$

Step 6.2.2

Combine $\frac{1}{3}$ and $- 343$ .

$- \frac{1}{4} (\frac{- 343}{3} - \frac{1}{3} \cdot 1^{3})$

Step 6.2.3

Move the negative in front of the fraction.

$- \frac{1}{4} (- \frac{343}{3} - \frac{1}{3} \cdot 1^{3})$

Step 6.2.4

One to any power is one.

$- \frac{1}{4} (- \frac{343}{3} - \frac{1}{3} \cdot 1)$

Step 6.2.5

Multiply $- 1$ by $1$ .

$- \frac{1}{4} (- \frac{343}{3} - \frac{1}{3})$

Step 6.2.6

Combine the numerators over the common denominator.

$- \frac{1}{4} \cdot \frac{- 343 - 1}{3}$

Step 6.2.7

Subtract $1$ from $- 343$ .

$- \frac{1}{4} \cdot \frac{- 344}{3}$

Step 6.2.8

Move the negative in front of the fraction.

$- \frac{1}{4} (- \frac{344}{3})$

Step 6.2.9

Multiply $- 1$ by $- 1$ .

$1 (\frac{1}{4}) \frac{344}{3}$

Step 6.2.10

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

$\frac{1}{4} \cdot \frac{344}{3}$

Step 6.2.11

Multiply $\frac{1}{4}$ by $\frac{344}{3}$ .

$\frac{344}{4 \cdot 3}$

Step 6.2.12

Multiply $4$ by $3$ .

$\frac{344}{12}$

Step 6.2.13

Cancel the common factor of $344$ and $12$ .

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Step 6.2.13.1

Factor $4$ out of $344$ .

$\frac{4 (86)}{12}$

Step 6.2.13.2

Cancel the common factors.

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Step 6.2.13.2.1

Factor $4$ out of $12$ .

$\frac{4 \cdot 86}{4 \cdot 3}$

Step 6.2.13.2.2

Cancel the common factor.

$\frac{4 \cdot 86}{4 \cdot 3}$

Step 6.2.13.2.3

Rewrite the expression.

$\frac{86}{3}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$\frac{86}{3}$

Decimal Form:

$28.\overline{6}$

Mixed Number Form:

$28 \frac{2}{3}$

Step 8