Calculus Examples

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

Step 1

Separate the variables.

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Step 1.1

Regroup factors.

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

Step 1.2

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

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

Step 1.3

Simplify.

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Step 1.3.1

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

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

Step 1.3.2

Cancel the common factor of $1 + 2 y^{2}$ .

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Step 1.3.2.1

Cancel the common factor.

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

Step 1.3.2.2

Rewrite the expression.

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

Step 1.3.3

Cancel the common factor of $y$ .

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Step 1.3.3.1

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

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

Step 1.3.3.2

Cancel the common factor.

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

Step 1.3.3.3

Rewrite the expression.

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

Step 1.4

Rewrite the equation.

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

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

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

Step 2.2

Integrate the left side.

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Step 2.2.1

Split the fraction into multiple fractions.

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

Step 2.2.2

Split the single integral into multiple integrals.

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

Step 2.2.3

Cancel the common factor of $y^{2}$ and $y$ .

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Step 2.2.3.1

Factor $y$ out of $2 y^{2}$ .

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

Step 2.2.3.2

Cancel the common factors.

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Step 2.2.3.2.1

Raise $y$ to the power of $1$ .

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

Step 2.2.3.2.2

Factor $y$ out of $y^{1}$ .

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

Step 2.2.3.2.3

Cancel the common factor.

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

Step 2.2.3.2.4

Rewrite the expression.

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

Step 2.2.3.2.5

Divide $2 y$ by $1$ .

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

Step 2.2.4

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

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

Step 2.2.5

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

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

Step 2.2.6

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

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

Step 2.2.7

Simplify.

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Step 2.2.7.1

Simplify.

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

Step 2.2.7.2

Simplify.

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Step 2.2.7.2.1

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

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

Step 2.2.7.2.2

Cancel the common factor of $2$ .

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Step 2.2.7.2.2.1

Cancel the common factor.

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

Step 2.2.7.2.2.2

Rewrite the expression.

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

Step 2.2.7.2.3

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

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

Step 2.2.8

Reorder terms.

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

Step 2.3

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

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

Step 2.4

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

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