Calculus Examples

$\int_{1}^{2} {(4 - 3 x)}^{8} d x$

Step 1

Let $u = 4 - 3 x$ . Then $d u = - 3 d x$ , so $- \frac{1}{3} d u = d x$ . Rewrite using $u$ and $d$ $u$ .

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Step 1.1

Let $u = 4 - 3 x$ . Find $\frac{d u}{d x}$ .

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Step 1.1.1

Differentiate $4 - 3 x$ .

$\frac{d}{d x} [4 - 3 x]$

Step 1.1.2

Differentiate.

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Step 1.1.2.1

By the Sum Rule, the derivative of $4 - 3 x$ with respect to $x$ is $\frac{d}{d x} [4] + \frac{d}{d x} [- 3 x]$ .

$\frac{d}{d x} [4] + \frac{d}{d x} [- 3 x]$

Step 1.1.2.2

Since $4$ is constant with respect to $x$ , the derivative of $4$ with respect to $x$ is $0$ .

$0 + \frac{d}{d x} [- 3 x]$

Step 1.1.3

Evaluate $\frac{d}{d x} [- 3 x]$ .

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Step 1.1.3.1

Since $- 3$ is constant with respect to $x$ , the derivative of $- 3 x$ with respect to $x$ is $- 3 \frac{d}{d x} [x]$ .

$0 - 3 \frac{d}{d x} [x]$

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 $n = 1$ .

$0 - 3 \cdot 1$

Step 1.1.3.3

Multiply $- 3$ by $1$ .

$0 - 3$

Step 1.1.4

Subtract $3$ from $0$ .

$- 3$

Step 1.2

Substitute the lower limit in for $x$ in $u = 4 - 3 x$ .

$u_{lower} = 4 - 3 \cdot 1$

Step 1.3

Simplify.

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Step 1.3.1

Multiply $- 3$ by $1$ .

$u_{lower} = 4 - 3$

Step 1.3.2

Subtract $3$ from $4$ .

$u_{lower} = 1$

Step 1.4

Substitute the upper limit in for $x$ in $u = 4 - 3 x$ .

$u_{upper} = 4 - 3 \cdot 2$

Step 1.5

Simplify.

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Step 1.5.1

Multiply $- 3$ by $2$ .

$u_{upper} = 4 - 6$

Step 1.5.2

Subtract $6$ from $4$ .

$u_{upper} = - 2$

Step 1.6

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

$u_{lower} = 1$

$u_{upper} = - 2$

Step 1.7

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

$\int_{1}^{- 2} u^{8} \frac{1}{- 3} d u$

Step 2

Simplify.

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Step 2.1

Move the negative in front of the fraction.

$\int_{1}^{- 2} u^{8} (- \frac{1}{3}) d u$

Step 2.2

Combine $u^{8}$ and $\frac{1}{3}$ .

$\int_{1}^{- 2} - \frac{u^{8}}{3} d u$

Step 3

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

$- \int_{1}^{- 2} \frac{u^{8}}{3} d u$

Step 4

Since $\frac{1}{3}$ is constant with respect to $u$ , move $\frac{1}{3}$ out of the integral.

$- (\frac{1}{3} \int_{1}^{- 2} u^{8} d u)$

Step 5

By the Power Rule, the integral of $u^{8}$ with respect to $u$ is $\frac{1}{9} u^{9}$ .

$- \frac{1}{3} \frac{1}{9} u^{9}]_{1}^{- 2}$

Step 6

Substitute and simplify.

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Step 6.1

Evaluate $\frac{1}{9} u^{9}$ at $- 2$ and at $1$ .

$- \frac{1}{3} ((\frac{1}{9} {(- 2)}^{9}) - \frac{1}{9} \cdot 1^{9})$

Step 6.2

Simplify.

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Step 6.2.1

Raise $- 2$ to the power of $9$ .

$- \frac{1}{3} (\frac{1}{9} \cdot - 512 - \frac{1}{9} \cdot 1^{9})$

Step 6.2.2

Combine $\frac{1}{9}$ and $- 512$ .

$- \frac{1}{3} (\frac{- 512}{9} - \frac{1}{9} \cdot 1^{9})$

Step 6.2.3

Move the negative in front of the fraction.

$- \frac{1}{3} (- \frac{512}{9} - \frac{1}{9} \cdot 1^{9})$

Step 6.2.4

One to any power is one.

$- \frac{1}{3} (- \frac{512}{9} - \frac{1}{9} \cdot 1)$

Step 6.2.5

Multiply $- 1$ by $1$ .

$- \frac{1}{3} (- \frac{512}{9} - \frac{1}{9})$

Step 6.2.6

Combine the numerators over the common denominator.

$- \frac{1}{3} \cdot \frac{- 512 - 1}{9}$

Step 6.2.7

Subtract $1$ from $- 512$ .

$- \frac{1}{3} \cdot \frac{- 513}{9}$

Step 6.2.8

Cancel the common factor of $- 513$ and $9$ .

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Step 6.2.8.1

Factor $9$ out of $- 513$ .

$- \frac{1}{3} \cdot \frac{9 \cdot - 57}{9}$

Step 6.2.8.2

Cancel the common factors.

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Step 6.2.8.2.1

Factor $9$ out of $9$ .

$- \frac{1}{3} \cdot \frac{9 \cdot - 57}{9 (1)}$

Step 6.2.8.2.2

Cancel the common factor.

$- \frac{1}{3} \cdot \frac{9 \cdot - 57}{9 \cdot 1}$

Step 6.2.8.2.3

Rewrite the expression.

$- \frac{1}{3} \cdot \frac{- 57}{1}$

Step 6.2.8.2.4

Divide $- 57$ by $1$ .

$- \frac{1}{3} \cdot - 57$

Step 6.2.9

Multiply $- 57$ by $- 1$ .

$57 (\frac{1}{3})$

Step 6.2.10

Combine $57$ and $\frac{1}{3}$ .

$\frac{57}{3}$

Step 6.2.11

Cancel the common factor of $57$ and $3$ .

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Step 6.2.11.1

Factor $3$ out of $57$ .

$\frac{3 \cdot 19}{3}$

Step 6.2.11.2

Cancel the common factors.

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Step 6.2.11.2.1

Factor $3$ out of $3$ .

$\frac{3 \cdot 19}{3 (1)}$

Step 6.2.11.2.2

Cancel the common factor.

$\frac{3 \cdot 19}{3 \cdot 1}$

Step 6.2.11.2.3

Rewrite the expression.

$\frac{19}{1}$

Step 6.2.11.2.4

Divide $19$ by $1$ .

$19$

Step 7