Calculus Examples

$\frac{d x}{d y} (6 x - 2 x y) = 1$

Step 1

Separate the variables.

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Step 1.1

Divide each term in $\frac{d x}{d y} (6 x - 2 x y) = 1$ by $6 x - 2 x y$ and simplify.

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Step 1.1.1

Divide each term in $\frac{d x}{d y} (6 x - 2 x y) = 1$ by $6 x - 2 x y$ .

$\frac{\frac{d x}{d y} (6 x - 2 x y)}{6 x - 2 x y} = \frac{1}{6 x - 2 x y}$

Step 1.1.2

Simplify the left side.

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Step 1.1.2.1

Cancel the common factor of $6 x - 2 x y$ .

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Step 1.1.2.1.1

Cancel the common factor.

$\frac{\frac{d x}{d y} (6 x - 2 x y)}{6 x - 2 x y} = \frac{1}{6 x - 2 x y}$

Step 1.1.2.1.2

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

$\frac{d x}{d y} = \frac{1}{6 x - 2 x y}$

Step 1.1.3

Simplify the right side.

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Step 1.1.3.1

Simplify the denominator.

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Step 1.1.3.1.1

Factor $2 x$ out of $6 x - 2 x y$ .

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Step 1.1.3.1.1.1

Factor $2 x$ out of $6 x$ .

$\frac{d x}{d y} = \frac{1}{2 x (3) - 2 x y}$

Step 1.1.3.1.1.2

Factor $2 x$ out of $- 2 x y$ .

$\frac{d x}{d y} = \frac{1}{2 x (3) + 2 x (- 1 y)}$

Step 1.1.3.1.1.3

Factor $2 x$ out of $2 x (3) + 2 x (- 1 y)$ .

$\frac{d x}{d y} = \frac{1}{2 x (3 - 1 y)}$

Step 1.1.3.1.2

Rewrite $- 1 y$ as $- y$ .

$\frac{d x}{d y} = \frac{1}{2 x (3 - y)}$

Step 1.2

Regroup factors.

$\frac{d x}{d y} = \frac{1}{2 x} \cdot \frac{1}{3 - y}$

Step 1.3

Multiply both sides by $2 x$ .

$2 x \frac{d x}{d y} = 2 x (\frac{1}{2 x} \cdot \frac{1}{3 - y})$

Step 1.4

Simplify.

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Step 1.4.1

Multiply $\frac{1}{2 x}$ by $\frac{1}{3 - y}$ .

$2 x \frac{d x}{d y} = 2 x \frac{1}{2 x (3 - y)}$

Step 1.4.2

Cancel the common factor of $2 x$ .

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Step 1.4.2.1

Cancel the common factor.

$2 x \frac{d x}{d y} = 2 x \frac{1}{2 x (3 - y)}$

Step 1.4.2.2

Rewrite the expression.

$2 x \frac{d x}{d y} = \frac{1}{3 - y}$

Step 1.5

Rewrite the equation.

$2 x d x = \frac{1}{3 - y} d y$

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

$\int 2 x d x = \int \frac{1}{3 - y} d y$

Step 2.2

Integrate the left side.

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Step 2.2.1

Since $2$ is constant with respect to $x$ , move $2$ out of the integral.

$2 \int x d x = \int \frac{1}{3 - y} d y$

Step 2.2.2

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

$2 (\frac{1}{2} x^{2} + C_{1}) = \int \frac{1}{3 - y} d y$

Step 2.2.3

Simplify the answer.

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Step 2.2.3.1

Rewrite $2 (\frac{1}{2} x^{2} + C_{1})$ as $2 (\frac{1}{2}) x^{2} + C_{1}$ .

$2 (\frac{1}{2}) x^{2} + C_{1} = \int \frac{1}{3 - y} d y$

Step 2.2.3.2

Simplify.

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Step 2.2.3.2.1

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

$\frac{2}{2} x^{2} + C_{1} = \int \frac{1}{3 - y} d y$

Step 2.2.3.2.2

Cancel the common factor of $2$ .

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Step 2.2.3.2.2.1

Cancel the common factor.

$\frac{2}{2} x^{2} + C_{1} = \int \frac{1}{3 - y} d y$

Step 2.2.3.2.2.2

Rewrite the expression.

$1 x^{2} + C_{1} = \int \frac{1}{3 - y} d y$

Step 2.2.3.2.3

Multiply $x^{2}$ by $1$ .

$x^{2} + C_{1} = \int \frac{1}{3 - y} d y$

Step 2.3

Integrate the right side.

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Step 2.3.1

Let $u = 3 - y$ . Then $d u = - d y$ , so $- d u = d y$ . Rewrite using $u$ and $d$ $u$ .

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Step 2.3.1.1

Let $u = 3 - y$ . Find $\frac{d u}{d y}$ .

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Step 2.3.1.1.1

Rewrite.

$\frac{- 1}{1}$

Step 2.3.1.1.2

Divide $- 1$ by $1$ .

$- 1$

Step 2.3.1.2

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

$x^{2} + C_{1} = \int \frac{- 1}{u} d u$

Step 2.3.2

Split the fraction into multiple fractions.

$x^{2} + C_{1} = \int - \frac{1}{u} d u$

Step 2.3.3

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

$x^{2} + C_{1} = - \int \frac{1}{u} d u$

Step 2.3.4

The integral of $\frac{1}{u}$ with respect to $u$ is $\ln (| u |)$ .

$x^{2} + C_{1} = - (\ln (| u |) + C_{2})$

Step 2.3.5

Simplify.

$x^{2} + C_{1} = - \ln (| u |) + C_{2}$

Step 2.3.6

Replace all occurrences of $u$ with $3 - y$ .

$x^{2} + C_{1} = - \ln (| 3 - y |) + C_{2}$

Step 2.4

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

$x^{2} = - \ln (| 3 - y |) + C$

Step 3

Solve for $x$ .

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Step 3.1

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

$x = \pm \sqrt{- \ln (| 3 - y |) + C}$

Step 3.2

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

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Step 3.2.1

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

$x = \sqrt{- \ln (| 3 - y |) + C}$

Step 3.2.2

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

$x = - \sqrt{- \ln (| 3 - y |) + C}$

Step 3.2.3

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

$x = \sqrt{- \ln (| 3 - y |) + C}$

$x = - \sqrt{- \ln (| 3 - y |) + C}$

$x = \sqrt{- \ln (| 3 - y |) + C}$

$x = - \sqrt{- \ln (| 3 - y |) + C}$

$x = \sqrt{- \ln (| 3 - y |) + C}$

$x = - \sqrt{- \ln (| 3 - y |) + C}$