Calculus Examples

$\frac{d y}{d x} = \frac{6 x^{2}}{2 y + \sin (y)}$

Step 1

Separate the variables.

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Step 1.1

Multiply both sides by $2 y + \sin (y)$ .

$(2 y + \sin (y)) \frac{d y}{d x} = (2 y + \sin (y)) \frac{6 x^{2}}{2 y + \sin (y)}$

Step 1.2

Cancel the common factor of $2 y + \sin (y)$ .

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Step 1.2.1

Cancel the common factor.

$(2 y + \sin (y)) \frac{d y}{d x} = (2 y + \sin (y)) \frac{6 x^{2}}{2 y + \sin (y)}$

Step 1.2.2

Rewrite the expression.

$(2 y + \sin (y)) \frac{d y}{d x} = 6 x^{2}$

Step 1.3

Rewrite the equation.

$(2 y + \sin (y)) d y = 6 x^{2} d x$

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

$\int 2 y + \sin (y) d y = \int 6 x^{2} d x$

Step 2.2

Integrate the left side.

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Step 2.2.1

Split the single integral into multiple integrals.

$\int 2 y d y + \int \sin (y) d y = \int 6 x^{2} d x$

Step 2.2.2

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

$2 \int y d y + \int \sin (y) d y = \int 6 x^{2} d x$

Step 2.2.3

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

$2 (\frac{1}{2} y^{2} + C_{1}) + \int \sin (y) d y = \int 6 x^{2} d x$

Step 2.2.4

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

$2 (\frac{1}{2} y^{2} + C_{1}) - \cos (y) + C_{2} = \int 6 x^{2} d x$

Step 2.2.5

Simplify.

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Step 2.2.5.1

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

$2 (\frac{y^{2}}{2} + C_{1}) - \cos (y) + C_{2} = \int 6 x^{2} d x$

Step 2.2.5.2

Simplify.

$y^{2} - \cos (y) + C_{3} = \int 6 x^{2} d x$

Step 2.3

Integrate the right side.

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Step 2.3.1

Since $6$ is constant with respect to $x$ , move $6$ out of the integral.

$y^{2} - \cos (y) + C_{3} = 6 \int x^{2} d x$

Step 2.3.2

By the Power Rule, the integral of $x^{2}$ with respect to $x$ is $\frac{1}{3} x^{3}$ .

$y^{2} - \cos (y) + C_{3} = 6 (\frac{1}{3} x^{3} + C_{4})$

Step 2.3.3

Simplify the answer.

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Step 2.3.3.1

Rewrite $6 (\frac{1}{3} x^{3} + C_{4})$ as $6 (\frac{1}{3}) x^{3} + C_{4}$ .

$y^{2} - \cos (y) + C_{3} = 6 (\frac{1}{3}) x^{3} + C_{4}$

Step 2.3.3.2

Simplify.

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Step 2.3.3.2.1

Combine $6$ and $\frac{1}{3}$ .

$y^{2} - \cos (y) + C_{3} = \frac{6}{3} x^{3} + C_{4}$

Step 2.3.3.2.2

Cancel the common factor of $6$ and $3$ .

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Step 2.3.3.2.2.1

Factor $3$ out of $6$ .

$y^{2} - \cos (y) + C_{3} = \frac{3 \cdot 2}{3} x^{3} + C_{4}$

Step 2.3.3.2.2.2

Cancel the common factors.

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Step 2.3.3.2.2.2.1

Factor $3$ out of $3$ .

$y^{2} - \cos (y) + C_{3} = \frac{3 \cdot 2}{3 (1)} x^{3} + C_{4}$

Step 2.3.3.2.2.2.2

Cancel the common factor.

$y^{2} - \cos (y) + C_{3} = \frac{3 \cdot 2}{3 \cdot 1} x^{3} + C_{4}$

Step 2.3.3.2.2.2.3

Rewrite the expression.

$y^{2} - \cos (y) + C_{3} = \frac{2}{1} x^{3} + C_{4}$

Step 2.3.3.2.2.2.4

Divide $2$ by $1$ .

$y^{2} - \cos (y) + C_{3} = 2 x^{3} + C_{4}$

Step 2.4

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

$y^{2} - \cos (y) = 2 x^{3} + K$