Calculus Examples

$\int x^{\frac{1}{3}} {(28 - x)}^{2} d x$

Step 1

Let $u_{1} = 28 - x$ . Then $d u_{1} = - d x$ , so $- d u_{1} = d x$ . Rewrite using $u_{1}$ and $d$ $u_{1}$ .

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Step 1.1

Let $u_{1} = 28 - x$ . Find $\frac{d u_{1}}{d x}$ .

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Step 1.1.1

Differentiate $28 - x$ .

$\frac{d}{d x} [28 - x]$

Step 1.1.2

Differentiate.

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Step 1.1.2.1

By the Sum Rule, the derivative of $28 - x$ with respect to $x$ is $\frac{d}{d x} [28] + \frac{d}{d x} [- x]$ .

$\frac{d}{d x} [28] + \frac{d}{d x} [- x]$

Step 1.1.2.2

Since $28$ is constant with respect to $x$ , the derivative of $28$ with respect to $x$ is $0$ .

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

Step 1.1.3

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

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Step 1.1.3.1

Since $- 1$ is constant with respect to $x$ , the derivative of $- x$ with respect to $x$ is $- \frac{d}{d x} [x]$ .

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

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 $n = 1$ .

$0 - 1 \cdot 1$

Step 1.1.3.3

Multiply $- 1$ by $1$ .

$0 - 1$

Step 1.1.4

Subtract $1$ from $0$ .

$- 1$

Step 1.2

Rewrite the problem using $u_{1}$ and $d u_{1}$ .

$\int - {(- u_{1} + 28)}^{\frac{1}{3}} {u_{1}}^{2} d u_{1}$

Step 2

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

$- \int {(- u_{1} + 28)}^{\frac{1}{3}} {u_{1}}^{2} d u_{1}$

Step 3

Let $u_{2} = - u_{1} + 28$ . Then $d u_{2} = - d u_{1}$ , so $- d u_{2} = d u_{1}$ . Rewrite using $u_{2}$ and $d$ $u_{2}$ .

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Step 3.1

Let $u_{2} = - u_{1} + 28$ . Find $\frac{d u_{2}}{d u_{1}}$ .

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Step 3.1.1

Differentiate $- u_{1} + 28$ .

$\frac{d}{d u_{1}} [- u_{1} + 28]$

Step 3.1.2

By the Sum Rule, the derivative of $- u_{1} + 28$ with respect to $u_{1}$ is $\frac{d}{d u_{1}} [- u_{1}] + \frac{d}{d u_{1}} [28]$ .

$\frac{d}{d u_{1}} [- u_{1}] + \frac{d}{d u_{1}} [28]$

Step 3.1.3

Evaluate $\frac{d}{d u_{1}} [- u_{1}]$ .

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Step 3.1.3.1

Since $- 1$ is constant with respect to $u_{1}$ , the derivative of $- u_{1}$ with respect to $u_{1}$ is $- \frac{d}{d u_{1}} [u_{1}]$ .

$- \frac{d}{d u_{1}} [u_{1}] + \frac{d}{d u_{1}} [28]$

Step 3.1.3.2

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

$- 1 \cdot 1 + \frac{d}{d u_{1}} [28]$

Step 3.1.3.3

Multiply $- 1$ by $1$ .

$- 1 + \frac{d}{d u_{1}} [28]$

Step 3.1.4

Differentiate using the Constant Rule.

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Step 3.1.4.1

Since $28$ is constant with respect to $u_{1}$ , the derivative of $28$ with respect to $u_{1}$ is $0$ .

$- 1 + 0$

Step 3.1.4.2

Add $- 1$ and $0$ .

$- 1$

Step 3.2

Rewrite the problem using $u_{2}$ and $d u_{2}$ .

$- \int - {u_{2}}^{\frac{1}{3}} {(- u_{2} + 28)}^{2} d u_{2}$

Step 4

Since $- 1$ is constant with respect to $u_{2}$ , move $- 1$ out of the integral.

$- - \int {u_{2}}^{\frac{1}{3}} {(- u_{2} + 28)}^{2} d u_{2}$

Step 5

Simplify.

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Step 5.1

Multiply $- 1$ by $- 1$ .

$1 \int {u_{2}}^{\frac{1}{3}} {(- u_{2} + 28)}^{2} d u_{2}$

Step 5.2

Multiply $\int {u_{2}}^{\frac{1}{3}} {(- u_{2} + 28)}^{2} d u_{2}$ by $1$ .

$\int {u_{2}}^{\frac{1}{3}} {(- u_{2} + 28)}^{2} d u_{2}$

Step 6

Simplify.

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Step 6.1

Rewrite ${(- u_{2} + 28)}^{2}$ as $(- u_{2} + 28) (- u_{2} + 28)$ .

$\int {u_{2}}^{\frac{1}{3}} ((- u_{2} + 28) (- u_{2} + 28)) d u_{2}$

Step 6.2