Calculus Examples

$y \sin^{2} (x) d y - (y d) x = 0$

Step 1

Add $y d x$ to both sides of the equation.

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

Step 2

Multiply both sides by $\frac{1}{\sin^{2} (x) y}$ .

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

Step 3

Simplify.

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Step 3.1

Cancel the common factor of $y \sin^{2} (x)$ .

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Step 3.1.1

Factor $y \sin^{2} (x)$ out of $\sin^{2} (x) y$ .

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

Step 3.1.2

Cancel the common factor.

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

Step 3.1.3

Rewrite the expression.

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

Step 3.2

Cancel the common factor of $y$ .

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Step 3.2.1

Factor $y$ out of $\sin^{2} (x) y$ .

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

Step 3.2.2

Cancel the common factor.

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

Step 3.2.3

Rewrite the expression.

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

Step 3.3

Rewrite $1$ as $1^{2}$ .

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

Step 3.4

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

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

Step 3.5

Convert from $\frac{1}{\sin (x)}$ to $\csc (x)$ .

$1 d y = \csc^{2} (x) d x$

Step 4

Integrate both sides.

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Step 4.1

Set up an integral on each side.

$\int d y = \int \csc^{2} (x) d x$

Step 4.2

Apply the constant rule.

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

Step 4.3

Since the derivative of $- \cot (x)$ is $\csc^{2} (x)$ , the integral of $\csc^{2} (x)$ is $- \cot (x)$ .

$y + C_{1} = - \cot (x) + C_{2}$

Step 4.4

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

$y = - \cot (x) + K$