Calculus Examples

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

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

Step 1

Separate the variables.

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Step 1.1

Multiply both sides by

\tan^{2} (y)

$\tan^{2} (y)$ .

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

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

Step 1.2

Cancel the common factor of

\tan^{2} (y)

$\tan^{2} (y)$ .

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Step 1.2.1

Cancel the common factor.

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

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

Step 1.2.2

Rewrite the expression.

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

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

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

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

Step 1.3

Rewrite the equation.

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

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

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

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

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

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

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

Step 2.2

Integrate the left side.

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Step 2.2.1

Using the Pythagorean Identity, rewrite

\tan^{2} (y)

$\tan^{2} (y)$ as

- 1 + \sec^{2} (y)

$- 1 + \sec^{2} (y)$ .

\int - 1 + \sec^{2} (y) d y = \int \sec^{2} (x) d x

$\int - 1 + \sec^{2} (y) d y = \int \sec^{2} (x) d x$

Step 2.2.2

Split the single integral into multiple integrals.

\int - 1 d y + \int \sec^{2} (y) d y = \int \sec^{2} (x) d x

$\int - 1 d y + \int \sec^{2} (y) d y = \int \sec^{2} (x) d x$

Step 2.2.3

Apply the constant rule.

- y + C_{1} + \int \sec^{2} (y) d y = \int \sec^{2} (x) d x

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

Step 2.2.4

Since the derivative of

\tan (y)

$\tan (y)$ is

\sec^{2} (y)

$\sec^{2} (y)$ , the integral of

\sec^{2} (y)

$\sec^{2} (y)$ is

\tan (y)

$\tan (y)$ .

- y + C_{1} + \tan (y) + C_{2} = \int \sec^{2} (x) d x

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

Step 2.2.5

Simplify.

- y + \tan (y) + C_{3} = \int \sec^{2} (x) d x

$- y + \tan (y) + C_{3} = \int \sec^{2} (x) d x$

- y + \tan (y) + C_{3} = \int \sec^{2} (x) d x

$- y + \tan (y) + C_{3} = \int \sec^{2} (x) d x$

Step 2.3

Since the derivative of

\tan (x)

$\tan (x)$ is

\sec^{2} (x)

$\sec^{2} (x)$ , the integral of

\sec^{2} (x)

$\sec^{2} (x)$ is

\tan (x)

$\tan (x)$ .

- y + \tan (y) + C_{3} = \tan (x) + C_{4}

$- y + \tan (y) + C_{3} = \tan (x) + C_{4}$

Step 2.4

Group the constant of integration on the right side as

K

$K$ .

- y + \tan (y) = \tan (x) + K

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

- y + \tan (y) = \tan (x) + K

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