Calculus Examples

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

Step 1

Separate the variables.

Tap for more steps...

Step 1.1

Divide each term in $\sin (x) \frac{d y}{d x} = \cos (x)$ by $\sin (x)$ and simplify.

Tap for more steps...

Step 1.1.1

Divide each term in $\sin (x) \frac{d y}{d x} = \cos (x)$ by $\sin (x)$ .

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

Step 1.1.2

Simplify the left side.

Tap for more steps...

Step 1.1.2.1

Cancel the common factor of $\sin (x)$ .

Tap for more steps...

Step 1.1.2.1.1

Cancel the common factor.

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

Step 1.1.2.1.2

Divide $\frac{d y}{d x}$ by $1$ .

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

Step 1.1.3

Simplify the right side.

Tap for more steps...

Step 1.1.3.1

Convert from $\frac{\cos (x)}{\sin (x)}$ to $\cot (x)$ .

$\frac{d y}{d x} = \cot (x)$

Step 1.2

Rewrite the equation.

$d y = \cot (x) d x$

Step 2

Integrate both sides.

Tap for more steps...

Step 2.1

Set up an integral on each side.

$\int d y = \int \cot (x) d x$

Step 2.2

Apply the constant rule.

$y + C_{1} = \int \cot (x) d x$

Step 2.3

The integral of $\cot (x)$ with respect to $x$ is $\ln (| \sin (x) |)$ .

$y + C_{1} = \ln (| \sin (x) |) + C_{2}$

Step 2.4

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

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