Calculus Examples

$\csc^{3} (x) \frac{d y}{d x} = 0$

Step 1

Separate the variables.

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Step 1.1

Divide each term in $\csc^{3} (x) \frac{d y}{d x} = 0$ by $\csc^{3} (x)$ and simplify.

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Step 1.1.1

Divide each term in $\csc^{3} (x) \frac{d y}{d x} = 0$ by $\csc^{3} (x)$ .

$\frac{\csc^{3} (x) \frac{d y}{d x}}{\csc^{3} (x)} = \frac{0}{\csc^{3} (x)}$

Step 1.1.2

Simplify the left side.

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Step 1.1.2.1

Cancel the common factor of $\csc^{3} (x)$ .

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Step 1.1.2.1.1

Cancel the common factor.

$\frac{\csc^{3} (x) \frac{d y}{d x}}{\csc^{3} (x)} = \frac{0}{\csc^{3} (x)}$

Step 1.1.2.1.2

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

$\frac{d y}{d x} = \frac{0}{\csc^{3} (x)}$

Step 1.1.3

Simplify the right side.

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Step 1.1.3.1

Divide $0$ by $\csc^{3} (x)$ .

$\frac{d y}{d x} = 0$

Step 1.2

Rewrite the equation.

$d y = 0 d x$

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

$\int d y = \int 0 d x$

Step 2.2

Apply the constant rule.

$y + C_{1} = \int 0 d x$

Step 2.3

Integrate the right side.

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Step 2.3.1

The integral of $0$ with respect to $x$ is $0$ .

$y + C_{1} = 0 + C_{2}$

Step 2.3.2

Add $0$ and $C_{2}$ .

$y + C_{1} = C_{2}$

Step 2.4

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

$y = D$