Calculus Examples

$\int x^{2} \sqrt{1 - x} d x$

Step 1

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

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Step 1.1.1

Differentiate $1 - x$ .

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

Step 1.1.2

Differentiate.

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Step 1.1.2.1

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

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

Step 1.1.2.2

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

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

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 $d u_{1}$ .

$\int - {(- u_{1} + 1)}^{2} \sqrt{u_{1}} d u_{1}$

Step 2

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

$- \int {(- u_{1} + 1)}^{2} \sqrt{u_{1}} d u_{1}$

Step 3

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{u_{1}}$ as ${u_{1}}^{\frac{1}{2}}$ .

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

Step 4

Let $u_{2} = - u_{1} + 1$ . 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 4.1

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

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Step 4.1.1

Differentiate $- u_{1} + 1$ .

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

Step 4.1.2

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

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

Step 4.1.3

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

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Step 4.1.3.1

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

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

Step 4.1.3.2

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

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

Step 4.1.3.3

Multiply $- 1$ by $1$ .

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

Step 4.1.4

Differentiate using the Constant Rule.

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Step 4.1.4.1

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

$- 1 + 0$

Step 4.1.4.2

Add $- 1$ and $0$ .

$- 1$

Step 4.2

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

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

Step 5

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

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

Step 6

Simplify.

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Step 6.1

Multiply $- 1$ by $- 1$ .

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

Step 6.2

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

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

Step 7

Let $u_{3} = - u_{2} + 1$ . Then $d u_{3} = - d u_{2}$ , so $- d u_{3} = d u_{2}$ . Rewrite using $u_{3}$ and $d$ $u_{3}$ .

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Step 7.1

Let $u_{3} = - u_{2} + 1$ . Find $\frac{d u_{3}}{d u_{2}}$ .

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Step 7.1.1

Differentiate $- u_{2} + 1$ .

$\frac{d}{d u_{2}} [- u_{2} + 1]$

Step 7.1.2

By the Sum Rule, the derivative of $- u_{2} + 1$ with respect to $u_{2}$ is $\frac{d}{d u_{2}} [- u_{2}] + \frac{d}{d u_{2}} [1]$ .

$\frac{d}{d u_{2}} [- u_{2}] + \frac{d}{d u_{2}} [1]$

Step 7.1.3

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

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Step 7.1.3.1

Since $- 1$ is constant with respect to $u_{2}$ , the derivative of $- u_{2}$ with respect to $u_{2}$ is $- \frac{d}{d u_{2}} [u_{2}]$ .

$- \frac{d}{d u_{2}} [u_{2}] + \frac{d}{d u_{2}} [1]$

Step 7.1.3.2

Differentiate using the Power Rule which states that $\frac{d}{d u_{2}} [{u_{2}}^{n}]$ is $n {u_{2}}^{n - 1}$ where $n = 1$ .

$- 1 \cdot 1 + \frac{d}{d u_{2}} [1]$

Step 7.1.3.3

Multiply $- 1$ by $1$ .

$- 1 + \frac{d}{d u_{2}} [1]$

Step 7.1.4

Differentiate using the Constant Rule.

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Step 7.1.4.1

Since $1$ is constant with respect to $u_{2}$ , the derivative of $1$ with respect to $u_{2}$ is $0$ .

$- 1 + 0$

Step 7.1.4.2

Add $- 1$ and $0$ .

$- 1$

Step 7.2

Rewrite the problem using $u_{3}$ and $d u_{3}$ .

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

Step 8

Since $- 1$ is constant with respect to $u_{3}$ , move $- 1$ out of the integral.

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

Step 9

Simplify.

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Step 9.1

Rewrite ${(- u_{3} + 1)}^{2}$ as $(- u_{3} + 1) (- u_{3} + 1)$ .

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

Step 9.2