Calculus Examples

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

Step 1

Rewrite the equation.

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

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

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

Step 2.2

Apply the constant rule.

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

Step 2.3

Integrate the right side.

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Step 2.3.1

Using the Pythagorean Identity, rewrite $\tan^{2} (x)$ as $- 1 + \sec^{2} (x)$ .

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

Step 2.3.2

Split the single integral into multiple integrals.

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

Step 2.3.3

Apply the constant rule.

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

Step 2.3.4

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

$y + C_{1} = - x + C_{2} + \tan (x) + C_{3}$

Step 2.3.5

Simplify.

$y + C_{1} = - x + \tan (x) + C_{4}$

Step 2.4

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

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