Calculus Examples

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

Step 1

Separate the variables.

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Step 1.1

Divide each term in $x \frac{d y}{d x} = \sec (y)$ by $x$ and simplify.

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Step 1.1.1

Divide each term in $x \frac{d y}{d x} = \sec (y)$ by $x$ .

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

Step 1.1.2

Simplify the left side.

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Step 1.1.2.1

Cancel the common factor of $x$ .

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Step 1.1.2.1.1

Cancel the common factor.

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

Step 1.1.2.1.2

Divide $\frac{d y}{d x}$ by $1$ .

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

Step 1.2

Multiply both sides by $\frac{1}{\sec (y)}$ .

$\frac{1}{\sec (y)} \frac{d y}{d x} = \frac{1}{\sec (y)} \cdot \frac{\sec (y)}{x}$

Step 1.3

Cancel the common factor of $\sec (y)$ .

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Step 1.3.1

Cancel the common factor.

$\frac{1}{\sec (y)} \frac{d y}{d x} = \frac{1}{\sec (y)} \cdot \frac{\sec (y)}{x}$

Step 1.3.2

Rewrite the expression.

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

Step 1.4

Rewrite the equation.

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

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

$\int \frac{1}{\sec (y)} d y = \int \frac{1}{x} d x$

Step 2.2

Integrate the left side.

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Step 2.2.1

Simplify.

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Step 2.2.1.1

Rewrite $\sec (y)$ in terms of sines and cosines.

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

Step 2.2.1.2

Multiply by the reciprocal of the fraction to divide by $\frac{1}{\cos (y)}$ .

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

Step 2.2.1.3

Multiply $\cos (y)$ by $1$ .

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

Step 2.2.2

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

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

Step 2.3

The integral of $\frac{1}{x}$ with respect to $x$ is $\ln (| x |)$ .

$\sin (y) + C_{1} = \ln (| x |) + C_{2}$

Step 2.4

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

$\sin (y) = \ln (| x |) + C$

Step 3

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

$y = \arcsin (\ln (| x |) + C)$