Calculus Examples

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

Step 1

Separate the variables.

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Step 1.1

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

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

Step 1.2

Simplify.

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Step 1.2.1

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

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Step 1.2.1.1

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

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

Step 1.2.1.2

Cancel the common factor.

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

Step 1.2.1.3

Rewrite the expression.

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

Step 1.2.2

Rewrite $\tan (x)$ in terms of sines and cosines.

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

Step 1.2.3

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

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

Step 1.3

Rewrite the equation.

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

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

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

Step 2.2

Integrate the left side.

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Step 2.2.1

Convert from $\frac{1}{\cos^{2} (y)}$ to $\sec^{2} (y)$ .

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

Step 2.2.2

Since the derivative of $\tan (y)$ is $\sec^{2} (y)$ , the integral of $\sec^{2} (y)$ is $\tan (y)$ .

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

Step 2.3

The integral of $\tan (x)$ with respect to $x$ is $\ln (| \sec (x) |)$ .

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

Step 2.4

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

$\tan (y) = \ln (| \sec (x) |) + C$

Step 3

Take the inverse tangent of both sides of the equation to extract $y$ from inside the tangent.

$y = \arctan (\ln (| \sec (x) |) + C)$