Calculus Examples

$\int_{1}^{2} \frac{16 {(3 - 2 x)}^{\frac{5}{3}}}{3} d x$

Step 1

Since $\frac{16}{3}$ is constant with respect to $x$ , move $\frac{16}{3}$ out of the integral.

$\frac{16}{3} \int_{1}^{2} {(3 - 2 x)}^{\frac{5}{3}} d x$

Step 2

Let $u = 3 - 2 x$ . Then $d u = - 2 d x$ , so $- \frac{1}{2} d u = d x$ . Rewrite using $u$ and $d$ $u$ .

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Step 2.1

Let $u = 3 - 2 x$ . Find $\frac{d u}{d x}$ .

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Step 2.1.1

Differentiate $3 - 2 x$ .

$\frac{d}{d x} [3 - 2 x]$

Step 2.1.2

Differentiate.

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Step 2.1.2.1

By the Sum Rule, the derivative of $3 - 2 x$ with respect to $x$ is $\frac{d}{d x} [3] + \frac{d}{d x} [- 2 x]$ .

$\frac{d}{d x} [3] + \frac{d}{d x} [- 2 x]$

Step 2.1.2.2

Since $3$ is constant with respect to $x$ , the derivative of $3$ with respect to $x$ is $0$ .

$0 + \frac{d}{d x} [- 2 x]$

Step 2.1.3

Evaluate $\frac{d}{d x} [- 2 x]$ .

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Step 2.1.3.1

Since $- 2$ is constant with respect to $x$ , the derivative of $- 2 x$ with respect to $x$ is $- 2 \frac{d}{d x} [x]$ .

$0 - 2 \frac{d}{d x} [x]$

Step 2.1.3.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 1$ .

$0 - 2 \cdot 1$

Step 2.1.3.3

Multiply $- 2$ by $1$ .

$0 - 2$

Step 2.1.4

Subtract $2$ from $0$ .

$- 2$

Step 2.2

Substitute the lower limit in for $x$ in $u = 3 - 2 x$ .

$u_{lower} = 3 - 2 \cdot 1$

Step 2.3

Simplify.

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Step 2.3.1

Multiply $- 2$ by $1$ .

$u_{lower} = 3 - 2$

Step 2.3.2

Subtract $2$ from $3$ .

$u_{lower} = 1$

Step 2.4

Substitute the upper limit in for $x$ in $u = 3 - 2 x$ .

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

Step 2.5

Simplify.

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Step 2.5.1

Multiply $- 2$ by $2$ .

$u_{upper} = 3 - 4$

Step 2.5.2

Subtract $4$ from $3$ .

$u_{upper} = - 1$

Step 2.6

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

$u_{lower} = 1$

$u_{upper} = - 1$

Step 2.7

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

$\frac{16}{3} \int_{1}^{- 1} u^{\frac{5}{3}} \frac{1}{- 2} d u$

Step 3

Simplify.

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Step 3.1

Move the negative in front of the fraction.

$\frac{16}{3} \int_{1}^{- 1} u^{\frac{5}{3}} (- \frac{1}{2}) d u$

Step 3.2

Combine $u^{\frac{5}{3}}$ and $\frac{1}{2}$ .

$\frac{16}{3} \int_{1}^{- 1} - \frac{u^{\frac{5}{3}}}{2} d u$

Step 4

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

$\frac{16}{3} (- \int_{1}^{- 1} \frac{u^{\frac{5}{3}}}{2} d u)$

Step 5

Since $\frac{1}{2}$ is constant with respect to $u$ , move $\frac{1}{2}$ out of the integral.

$\frac{16}{3} (- (\frac{1}{2} \int_{1}^{- 1} u^{\frac{5}{3}} d u))$

Step 6

Simplify.

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Step 6.1

Multiply $\frac{16}{3}$ by $\frac{1}{2}$ .

$- \frac{16}{3 \cdot 2} \int_{1}^{- 1} u^{\frac{5}{3}} d u$

Step 6.2

Multiply $3$ by $2$ .

$- \frac{16}{6} \int_{1}^{- 1} u^{\frac{5}{3}} d u$

Step 6.3

Cancel the common factor of $16$ and $6$ .

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Step 6.3.1

Factor $2$ out of $16$ .

$- \frac{2 (8)}{6} \int_{1}^{- 1} u^{\frac{5}{3}} d u$

Step 6.3.2

Cancel the common factors.

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Step 6.3.2.1

Factor $2$ out of $6$ .

$- \frac{2 \cdot 8}{2 \cdot 3} \int_{1}^{- 1} u^{\frac{5}{3}} d u$

Step 6.3.2.2

Cancel the common factor.

$- \frac{2 \cdot 8}{2 \cdot 3} \int_{1}^{- 1} u^{\frac{5}{3}} d u$

Step 6.3.2.3

Rewrite the expression.

$- \frac{8}{3} \int_{1}^{- 1} u^{\frac{5}{3}} d u$

Step 7

By the Power Rule, the integral of $u^{\frac{5}{3}}$ with respect to $u$ is $\frac{3}{8} u^{\frac{8}{3}}$ .

$- \frac{8}{3} \frac{3}{8} u^{\frac{8}{3}}]_{1}^{- 1}$

Step 8

Substitute and simplify.

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Step 8.1

Evaluate $\frac{3}{8} u^{\frac{8}{3}}$ at $- 1$ and at $1$ .

$- \frac{8}{3} ((\frac{3}{8} {(- 1)}^{\frac{8}{3}}) - \frac{3}{8} \cdot 1^{\frac{8}{3}})$

Step 8.2

Simplify.

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Step 8.2.1

Rewrite $- 1$ as ${(- 1)}^{3}$ .

$- \frac{8}{3} (\frac{3}{8} {({(- 1)}^{3})}^{\frac{8}{3}} - \frac{3}{8} \cdot 1^{\frac{8}{3}})$

Step 8.2.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$- \frac{8}{3} (\frac{3}{8} {(- 1)}^{3 (\frac{8}{3})} - \frac{3}{8} \cdot 1^{\frac{8}{3}})$

Step 8.2.3

Cancel the common factor of $3$ .

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Step 8.2.3.1

Cancel the common factor.

$- \frac{8}{3} (\frac{3}{8} {(- 1)}^{3 (\frac{8}{3})} - \frac{3}{8} \cdot 1^{\frac{8}{3}})$

Step 8.2.3.2

Rewrite the expression.

$- \frac{8}{3} (\frac{3}{8} {(- 1)}^{8} - \frac{3}{8} \cdot 1^{\frac{8}{3}})$

Step 8.2.4

Raise $- 1$ to the power of $8$ .

$- \frac{8}{3} (\frac{3}{8} \cdot 1 - \frac{3}{8} \cdot 1^{\frac{8}{3}})$

Step 8.2.5

Multiply $\frac{3}{8}$ by $1$ .

$- \frac{8}{3} (\frac{3}{8} - \frac{3}{8} \cdot 1^{\frac{8}{3}})$

Step 8.2.6

One to any power is one.

$- \frac{8}{3} (\frac{3}{8} - \frac{3}{8} \cdot 1)$

Step 8.2.7

Multiply $- 1$ by $1$ .

$- \frac{8}{3} (\frac{3}{8} - \frac{3}{8})$

Step 8.2.8

Combine the numerators over the common denominator.

$- \frac{8}{3} \cdot \frac{3 - 3}{8}$

Step 8.2.9

Subtract $3$ from $3$ .

$- \frac{8}{3} \cdot \frac{0}{8}$

Step 8.2.10

Cancel the common factor of $0$ and $8$ .

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Step 8.2.10.1

Factor $8$ out of $0$ .

$- \frac{8}{3} \cdot \frac{8 (0)}{8}$

Step 8.2.10.2

Cancel the common factors.

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Step 8.2.10.2.1

Factor $8$ out of $8$ .

$- \frac{8}{3} \cdot \frac{8 \cdot 0}{8 \cdot 1}$

Step 8.2.10.2.2

Cancel the common factor.

$- \frac{8}{3} \cdot \frac{8 \cdot 0}{8 \cdot 1}$

Step 8.2.10.2.3

Rewrite the expression.

$- \frac{8}{3} \cdot \frac{0}{1}$

Step 8.2.10.2.4

Divide $0$ by $1$ .

$- \frac{8}{3} \cdot 0$

Step 8.2.11

Multiply $0$ by $- 1$ .

$0 (\frac{8}{3})$

Step 8.2.12

Multiply $0$ by $\frac{8}{3}$ .

$0$

Step 9