Calculus Examples

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

Step 1

Separate the variables.

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Step 1.1

Factor $\frac{d y}{d x}$ out.

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

Step 1.2

Simplify $\frac{\cos^{2} (x)}{\cos^{2} (x)} \frac{d y}{d x}$ .

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Step 1.2.1

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

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Step 1.2.1.1

Cancel the common factor.

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

Step 1.2.1.2

Rewrite the expression.

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

Step 1.2.2

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

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

Step 1.3

Rewrite the equation.

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

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

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

Step 2.2

Apply the constant rule.

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

Step 2.3

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

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

Step 2.4

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

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