Calculus Examples

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

Step 1

Separate the variables.

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Step 1.1

Multiply both sides by $y + 2$ .

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

Step 1.2

Cancel the common factor of $y + 2$ .

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Step 1.2.1

Cancel the common factor.

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

Step 1.2.2

Rewrite the expression.

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

Step 1.3

Rewrite the equation.

$(y + 2) d y = \sin (x) d x$

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

$\int y + 2 d y = \int \sin (x) d x$

Step 2.2

Integrate the left side.

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Step 2.2.1

Split the single integral into multiple integrals.

$\int y d y + \int 2 d y = \int \sin (x) d x$

Step 2.2.2

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

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

Step 2.2.3

Apply the constant rule.

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

Step 2.2.4

Simplify.

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

Step 2.3

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

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

Step 2.4

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

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