Calculus Examples

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

Step 1

Separate the variables.

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Step 1.1

Multiply both sides by $\cos (y) + 1$ .

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

Step 1.2

Cancel the common factor of $\cos (y) + 1$ .

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Step 1.2.1

Cancel the common factor.

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

Step 1.2.2

Rewrite the expression.

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

Step 1.3

Rewrite the equation.

$(\cos (y) + 1) d y = 1 d x$

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

$\int \cos (y) + 1 d y = \int d x$

Step 2.2

Integrate the left side.

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Step 2.2.1

Split the single integral into multiple integrals.

$\int \cos (y) d y + \int d y = \int d x$

Step 2.2.2

The integral of $\cos (y)$ with respect to $y$ is $\sin (y)$ .

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

Step 2.2.3

Apply the constant rule.

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

Step 2.2.4

Simplify.

$\sin (y) + y + C_{3} = \int d x$

Step 2.2.5

Reorder terms.

$y + \sin (y) + C_{3} = \int d x$

Step 2.3

Apply the constant rule.

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

Step 2.4

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

$y + \sin (y) = x + K$