Calculus Examples

$2 x \cdot \frac{d y}{d x} = x \cdot \sin (2 x)$

Step 1

Separate the variables.

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Step 1.1

Divide each term in $2 x \cdot \frac{d y}{d x} = x \cdot \sin (2 x)$ by $2 x$ and simplify.

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Step 1.1.1

Divide each term in $2 x \cdot \frac{d y}{d x} = x \cdot \sin (2 x)$ by $2 x$ .

$\frac{2 x \cdot \frac{d y}{d x}}{2 x} = \frac{x \cdot \sin (2 x)}{2 x}$

Step 1.1.2

Simplify the left side.

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Step 1.1.2.1

Cancel the common factor of $2$ .

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Step 1.1.2.1.1

Cancel the common factor.

$\frac{2 x \cdot \frac{d y}{d x}}{2 x} = \frac{x \cdot \sin (2 x)}{2 x}$

Step 1.1.2.1.2

Rewrite the expression.

$\frac{x \cdot \frac{d y}{d x}}{x} = \frac{x \cdot \sin (2 x)}{2 x}$

Step 1.1.2.2

Cancel the common factor of $x$ .

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Step 1.1.2.2.1

Cancel the common factor.

$\frac{x \cdot \frac{d y}{d x}}{x} = \frac{x \cdot \sin (2 x)}{2 x}$

Step 1.1.2.2.2

Divide $\frac{d y}{d x}$ by $1$ .

$\frac{d y}{d x} = \frac{x \cdot \sin (2 x)}{2 x}$

Step 1.1.3

Simplify the right side.

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Step 1.1.3.1

Cancel the common factor of $x$ .

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Step 1.1.3.1.1

Cancel the common factor.

$\frac{d y}{d x} = \frac{x \cdot \sin (2 x)}{2 x}$

Step 1.1.3.1.2

Rewrite the expression.

$\frac{d y}{d x} = \frac{\sin (2 x)}{2}$

Step 1.2

Rewrite the equation.

$d y = \frac{\sin (2 x)}{2} d x$

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

$\int d y = \int \frac{\sin (2 x)}{2} d x$

Step 2.2

Apply the constant rule.

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

Step 2.3

Integrate the right side.

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Step 2.3.1

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

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

Step 2.3.2

Let $u = 2 x$ . 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 2.3.2.1

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

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Step 2.3.2.1.1

Differentiate $2 x$ .

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

Step 2.3.2.1.2

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

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

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 $n = 1$ .

$2 \cdot 1$

Step 2.3.2.1.4

Multiply $2$ by $1$ .

$2$

Step 2.3.2.2

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

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

Step 2.3.3

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

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

Step 2.3.4

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

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

Step 2.3.5

Simplify.

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Step 2.3.5.1

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

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

Step 2.3.5.2

Multiply $2$ by $2$ .

$y + C_{1} = \frac{1}{4} \int \sin (u) d u$

Step 2.3.6

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

$y + C_{1} = \frac{1}{4} (- \cos (u) + C_{2})$

Step 2.3.7

Simplify.

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Step 2.3.7.1

Simplify.

$y + C_{1} = \frac{1}{4} (- \cos (u)) + C_{2}$

Step 2.3.7.2

Combine $\frac{1}{4}$ and $\cos (u)$ .

$y + C_{1} = - \frac{\cos (u)}{4} + C_{2}$

Step 2.3.8

Replace all occurrences of $u$ with $2 x$ .

$y + C_{1} = - \frac{\cos (2 x)}{4} + C_{2}$

Step 2.3.9

Reorder terms.

$y + C_{1} = - \frac{1}{4} \cos (2 x) + C_{2}$

Step 2.4

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

$y = - \frac{1}{4} \cos (2 x) + K$