Finite Math Examples

$\int \sin^{2} (1 - x) d x$

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 $d u_{1}$ .

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

Step 2

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

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

Step 3

Use the half-angle formula to rewrite $\sin^{2} (u_{1})$ as $\frac{1 - \cos (2 u_{1})}{2}$ .

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

Step 4

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

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

Step 5

Split the single integral into multiple integrals.

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

Step 6

Apply the constant rule.

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

Step 7

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

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

Step 8

Let $u_{2} = 2 u_{1}$ . Then $d u_{2} = 2 d u_{1}$ , so $\frac{1}{2} d u_{2} = d u_{1}$ . Rewrite using $u_{2}$ and $d$ $u_{2}$ .

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Step 8.1

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

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Step 8.1.1

Differentiate $2 u_{1}$ .

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

Step 8.1.2

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

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

Step 8.1.3

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

$2 \cdot 1$

Step 8.1.4

Multiply $2$ by $1$ .

$2$

Step 8.2

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

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

Step 9

Combine $\cos (u_{2})$ and $\frac{1}{2}$ .

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

Step 10

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

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

Step 11

The integral of $\cos (u_{2})$ with respect to $u_{2}$ is $\sin (u_{2})$ .

$- \frac{1}{2} (u_{1} + C - \frac{1}{2} (\sin (u_{2}) + C))$

Step 12

Simplify.

$- \frac{1}{2} (u_{1} - \frac{1}{2} \sin (u_{2})) + C$

Step 13

Substitute back in for each integration substitution variable.

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Step 13.1

Replace all occurrences of $u_{1}$ with $1 - x$ .

$- \frac{1}{2} (1 - x - \frac{1}{2} \sin (u_{2})) + C$

Step 13.2

Replace all occurrences of $u_{2}$ with $2 u_{1}$ .

$- \frac{1}{2} (1 - x - \frac{1}{2} \sin (2 u_{1})) + C$

Step 13.3

Replace all occurrences of $u_{1}$ with $1 - x$ .

$- \frac{1}{2} (1 - x - \frac{1}{2} \sin (2 (1 - x))) + C$

Step 14

Simplify.

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Step 14.1

Simplify each term.

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Step 14.1.1

Apply the distributive property.

$- \frac{1}{2} (1 - x - \frac{1}{2} \sin (2 \cdot 1 + 2 (- x))) + C$

Step 14.1.2

Multiply $2$ by $1$ .

$- \frac{1}{2} (1 - x - \frac{1}{2} \sin (2 + 2 (- x))) + C$

Step 14.1.3

Multiply $- 1$ by $2$ .

$- \frac{1}{2} (1 - x - \frac{1}{2} \sin (2 - 2 x)) + C$

Step 14.1.4

Combine $\sin (2 - 2 x)$ and $\frac{1}{2}$ .

$- \frac{1}{2} (1 - x - \frac{\sin (2 - 2 x)}{2}) + C$

Step 14.2

Apply the distributive property.

$- \frac{1}{2} \cdot 1 - \frac{1}{2} (- x) - \frac{1}{2} (- \frac{\sin (2 - 2 x)}{2}) + C$

Step 14.3

Simplify.

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Step 14.3.1

Multiply $- 1$ by $1$ .

$- \frac{1}{2} - \frac{1}{2} (- x) - \frac{1}{2} (- \frac{\sin (2 - 2 x)}{2}) + C$

Step 14.3.2

Multiply $- \frac{1}{2} (- x)$ .

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Step 14.3.2.1

Multiply $- 1$ by $- 1$ .

$- \frac{1}{2} + 1 (\frac{1}{2}) x - \frac{1}{2} (- \frac{\sin (2 - 2 x)}{2}) + C$

Step 14.3.2.2

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

$- \frac{1}{2} + \frac{1}{2} x - \frac{1}{2} (- \frac{\sin (2 - 2 x)}{2}) + C$

Step 14.3.2.3

Combine $\frac{1}{2}$ and $x$ .

$- \frac{1}{2} + \frac{x}{2} - \frac{1}{2} (- \frac{\sin (2 - 2 x)}{2}) + C$

Step 14.3.3

Multiply $- \frac{1}{2} (- \frac{\sin (2 - 2 x)}{2})$ .

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Step 14.3.3.1

Multiply $- 1$ by $- 1$ .

$- \frac{1}{2} + \frac{x}{2} + 1 (\frac{1}{2}) \frac{\sin (2 - 2 x)}{2} + C$

Step 14.3.3.2

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

$- \frac{1}{2} + \frac{x}{2} + \frac{1}{2} \cdot \frac{\sin (2 - 2 x)}{2} + C$

Step 14.3.3.3

Multiply $\frac{1}{2}$ by $\frac{\sin (2 - 2 x)}{2}$ .

$- \frac{1}{2} + \frac{x}{2} + \frac{\sin (2 - 2 x)}{2 \cdot 2} + C$

Step 14.3.3.4

Multiply $2$ by $2$ .

$- \frac{1}{2} + \frac{x}{2} + \frac{\sin (2 - 2 x)}{4} + C$

Step 15

Reorder terms.

$- \frac{1}{2} + \frac{1}{2} x + \frac{1}{4} \sin (2 - 2 x) + C$