Calculus Examples

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

Step 1

Separate the variables.

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Step 1.1

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

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

Step 1.2

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

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Step 1.2.1

Cancel the common factor.

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

Step 1.2.2

Rewrite the expression.

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

Step 1.3

Remove unnecessary parentheses.

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

Step 1.4

Rewrite the equation.

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

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

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

Step 2.2

Integrate the left side.

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Step 2.2.1

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

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

Step 2.2.2

Since $D E$ is constant with respect to $y$ , move $D E$ out of the integral.

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

Step 2.2.3

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

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

Step 2.2.4

Simplify.

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

Step 2.3

Apply the constant rule.

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

Step 2.4

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

$D E \tan (y) = x + K$

Step 3

Solve for $y$ .

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Step 3.1

Divide each term in $D E \tan (y) = x + K$ by $D E$ and simplify.

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Step 3.1.1

Divide each term in $D E \tan (y) = x + K$ by $D E$ .

$\frac{D E \tan (y)}{D E} = \frac{x}{D E} + \frac{K}{D E}$

Step 3.1.2

Simplify the left side.

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Step 3.1.2.1

Cancel the common factor of $D$ .

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Step 3.1.2.1.1

Cancel the common factor.

$\frac{D E \tan (y)}{D E} = \frac{x}{D E} + \frac{K}{D E}$

Step 3.1.2.1.2

Rewrite the expression.

$\frac{E \tan (y)}{E} = \frac{x}{D E} + \frac{K}{D E}$

Step 3.1.2.2

Cancel the common factor of $E$ .

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Step 3.1.2.2.1

Cancel the common factor.

$\frac{E \tan (y)}{E} = \frac{x}{D E} + \frac{K}{D E}$

Step 3.1.2.2.2

Divide $\tan (y)$ by $1$ .

$\tan (y) = \frac{x}{D E} + \frac{K}{D E}$

Step 3.2

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

$y = \arctan (\frac{x}{D E} + \frac{K}{D E})$

Step 4

Simplify the constant of integration.

$y = \arctan (\frac{x}{K E} + \frac{K}{K E})$