Calculus Examples

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

Step 1

Separate the variables.

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Step 1.1

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

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

Step 1.2

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

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Step 1.2.1

Cancel the common factor.

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

Step 1.2.2

Rewrite the expression.

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

Step 1.3

Rewrite the equation.

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

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

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

Step 2.2

Integrate the left side.

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Step 2.2.1

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

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

Step 2.2.2

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

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

Step 2.3

Apply the constant rule.

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

Step 2.4

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

$\tan (y) = x + K$

Step 3

Solve for $y$ .

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Step 3.1

Reorder $x$ and $K$ .

$\tan (y) = K + x$

Step 3.2

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

$y = \arctan (K + x)$