Calculus Examples

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

Step 1

Separate the variables.

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Step 1.1

Regroup factors.

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

Step 1.2

Multiply both sides by $\frac{1 - 2 y}{y}$ .

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

Step 1.3

Simplify.

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Step 1.3.1

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

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

Step 1.3.2

Cancel the common factor of $1 - 2 y$ .

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Step 1.3.2.1

Cancel the common factor.

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

Step 1.3.2.2

Rewrite the expression.

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

Step 1.3.3

Cancel the common factor of $y$ .

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Step 1.3.3.1

Factor $y$ out of $y \cos (x)$ .

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

Step 1.3.3.2

Cancel the common factor.

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

Step 1.3.3.3

Rewrite the expression.

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

Step 1.4

Rewrite the equation.

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

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

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

Step 2.2

Integrate the left side.

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Step 2.2.1

Split the fraction into multiple fractions.

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

Step 2.2.2

Split the single integral into multiple integrals.

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

Step 2.2.3

Cancel the common factor of $y$ .

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Step 2.2.3.1

Cancel the common factor.

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

Step 2.2.3.2

Divide $- 2$ by $1$ .

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

Step 2.2.4

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

$\ln (| y |) + C_{1} + \int - 2 d y = \int \cos (x) d x$

Step 2.2.5

Apply the constant rule.

$\ln (| y |) + C_{1} - 2 y + C_{2} = \int \cos (x) d x$

Step 2.2.6

Simplify.

$\ln (| y |) - 2 y + C_{3} = \int \cos (x) d x$

Step 2.2.7

Reorder terms.

$- 2 y + \ln (| y |) + C_{3} = \int \cos (x) d x$

Step 2.3

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

$- 2 y + \ln (| y |) + C_{3} = \sin (x) + C_{4}$

Step 2.4

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

$- 2 y + \ln (| y |) = \sin (x) + C$