Calculus Examples

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

Step 1

Rewrite the equation.

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

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

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

Step 2.2

Apply the constant rule.

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

Step 2.3

Integrate the right side.

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Step 2.3.1

Convert from $\frac{1}{\sin^{2} (x)}$ to $\csc^{2} (x)$ .

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

Step 2.3.2

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 2.4

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

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