Calculus Examples

$\frac{d y}{d x} = - \frac{\sin (x + 5)}{y}$

Step 1

Separate the variables.

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Step 1.1

Multiply both sides by $y$ .

$y \frac{d y}{d x} = y (- \frac{\sin (x + 5)}{y})$

Step 1.2

Simplify.

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Step 1.2.1

Rewrite using the commutative property of multiplication.

$y \frac{d y}{d x} = - y \frac{\sin (x + 5)}{y}$

Step 1.2.2

Cancel the common factor of $y$ .

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Step 1.2.2.1

Factor $y$ out of $- y$ .

$y \frac{d y}{d x} = y \cdot - 1 \frac{\sin (x + 5)}{y}$

Step 1.2.2.2

Cancel the common factor.

$y \frac{d y}{d x} = y \cdot - 1 \frac{\sin (x + 5)}{y}$

Step 1.2.2.3

Rewrite the expression.

$y \frac{d y}{d x} = - \sin (x + 5)$

Step 1.3

Rewrite the equation.

$y d y = - \sin (x + 5) d x$

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

$\int y d y = \int - \sin (x + 5) d x$

Step 2.2

By the Power Rule, the integral of $y$ with respect to $y$ is $\frac{1}{2} y^{2}$ .

$\frac{1}{2} y^{2} + C_{1} = \int - \sin (x + 5) d x$

Step 2.3

Integrate the right side.

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Step 2.3.1

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

$\frac{1}{2} y^{2} + C_{1} = - \int \sin (x + 5) d x$

Step 2.3.2

Let $u = x + 5$ . Then $d u = d x$ . Rewrite using $u$ and $d$ $u$ .

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Step 2.3.2.1

Let $u = x + 5$ . Find $\frac{d u}{d x}$ .

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Step 2.3.2.1.1

Differentiate $x + 5$ .

$\frac{d}{d x} [x + 5]$

Step 2.3.2.1.2

By the Sum Rule, the derivative of $x + 5$ with respect to $x$ is $\frac{d}{d x} [x] + \frac{d}{d x} [5]$ .

$\frac{d}{d x} [x] + \frac{d}{d x} [5]$

Step 2.3.2.1.3

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

$1 + \frac{d}{d x} [5]$

Step 2.3.2.1.4

Since $5$ is constant with respect to $x$ , the derivative of $5$ with respect to $x$ is $0$ .

$1 + 0$

Step 2.3.2.1.5

Add $1$ and $0$ .

$1$

Step 2.3.2.2

Rewrite the problem using $u$ and $d u$ .

$\frac{1}{2} y^{2} + C_{1} = - \int \sin (u) d u$

Step 2.3.3

The integral of $\sin (u)$ with respect to $u$ is $- \cos (u)$ .

$\frac{1}{2} y^{2} + C_{1} = - (- \cos (u) + C_{2})$

Step 2.3.4

Simplify.

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Step 2.3.4.1

Simplify.

$\frac{1}{2} y^{2} + C_{1} = - - \cos (u) + C_{2}$

Step 2.3.4.2

Simplify.

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Step 2.3.4.2.1

Multiply $- 1$ by $- 1$ .

$\frac{1}{2} y^{2} + C_{1} = 1 \cos (u) + C_{2}$

Step 2.3.4.2.2

Multiply $\cos (u)$ by $1$ .

$\frac{1}{2} y^{2} + C_{1} = \cos (u) + C_{2}$

Step 2.3.5

Replace all occurrences of $u$ with $x + 5$ .

$\frac{1}{2} y^{2} + C_{1} = \cos (x + 5) + C_{2}$

Step 2.4

Group the constant of integration on the right side as $K$ .

$\frac{1}{2} y^{2} = \cos (x + 5) + K$

Step 3

Solve for $y$ .

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Step 3.1

Multiply both sides of the equation by $2$ .

$2 (\frac{1}{2} y^{2}) = 2 (\cos (x + 5) + K)$

Step 3.2

Simplify both sides of the equation.

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Step 3.2.1

Simplify the left side.

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Step 3.2.1.1

Simplify $2 (\frac{1}{2} y^{2})$ .

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Step 3.2.1.1.1

Combine $\frac{1}{2}$ and $y^{2}$ .

$2 \frac{y^{2}}{2} = 2 (\cos (x + 5) + K)$

Step 3.2.1.1.2

Cancel the common factor of $2$ .

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Step 3.2.1.1.2.1

Cancel the common factor.

$2 \frac{y^{2}}{2} = 2 (\cos (x + 5) + K)$

Step 3.2.1.1.2.2

Rewrite the expression.

$y^{2} = 2 (\cos (x + 5) + K)$

Step 3.2.2

Simplify the right side.

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Step 3.2.2.1

Apply the distributive property.

$y^{2} = 2 \cos (x + 5) + 2 K$

Step 3.3

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$y = \pm \sqrt{2 \cos (x + 5) + 2 K}$

Step 3.4

Factor $2$ out of $2 \cos (x + 5) + 2 K$ .

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Step 3.4.1

Factor $2$ out of $2 \cos (x + 5)$ .

$y = \pm \sqrt{2 (\cos (x + 5)) + 2 K}$

Step 3.4.2

Factor $2$ out of $2 K$ .

$y = \pm \sqrt{2 (\cos (x + 5)) + 2 K}$

Step 3.4.3

Factor $2$ out of $2 (\cos (x + 5)) + 2 K$ .

$y = \pm \sqrt{2 (\cos (x + 5) + K)}$

Step 3.5

The complete solution is the result of both the positive and negative portions of the solution.

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Step 3.5.1

First, use the positive value of the $\pm$ to find the first solution.

$y = \sqrt{2 (\cos (x + 5) + K)}$

Step 3.5.2

Next, use the negative value of the $\pm$ to find the second solution.

$y = - \sqrt{2 (\cos (x + 5) + K)}$

Step 3.5.3

The complete solution is the result of both the positive and negative portions of the solution.

$y = \sqrt{2 (\cos (x + 5) + K)}$

$y = - \sqrt{2 (\cos (x + 5) + K)}$

$y = \sqrt{2 (\cos (x + 5) + K)}$

$y = - \sqrt{2 (\cos (x + 5) + K)}$

$y = \sqrt{2 (\cos (x + 5) + K)}$

$y = - \sqrt{2 (\cos (x + 5) + K)}$