Calculus Examples

$\frac{d y}{d x} + y \tan (x) = 0$

Step 1

Separate the variables.

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Step 1.1

Subtract $y \tan (x)$ from both sides of the equation.

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

Step 1.2

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

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

Step 1.3

Simplify.

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Step 1.3.1

Rewrite using the commutative property of multiplication.

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

Step 1.3.2

Cancel the common factor of $y$ .

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Step 1.3.2.1

Move the leading negative in $- \frac{1}{y}$ into the numerator.

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

Step 1.3.2.2

Factor $y$ out of $y \tan (x)$ .

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

Step 1.3.2.3

Cancel the common factor.

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

Step 1.3.2.4

Rewrite the expression.

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

Step 1.4

Rewrite the equation.

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

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

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

Step 2.2

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

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

Step 2.3

Integrate the right side.

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Step 2.3.1

Since $- 1$ is constant with respect to $x$ , move $- 1$ out of the integral.

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

Step 2.3.2

The integral of $\tan (x)$ with respect to $x$ is $\ln (| \sec (x) |)$ .

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

Step 2.3.3

Simplify.

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

Step 2.4

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

$\ln (| y |) = - \ln (| \sec (x) |) + C$

Step 3

Solve for $y$ .

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Step 3.1

Move all the terms containing a logarithm to the left side of the equation.

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

Step 3.2

Use the product property of logarithms, $\log_{b} (x) + \log_{b} (y) = \log_{b} (x y)$ .

$\ln (| y | | \sec (x) |) = C$

Step 3.3

To multiply absolute values, multiply the terms inside each absolute value.

$\ln (| y \sec (x) |) = C$

Step 3.4

To solve for $y$ , rewrite the equation using properties of logarithms.

$e^{\ln (| y \sec (x) |)} = e^{C}$

Step 3.5

Rewrite $\ln (| y \sec (x) |) = C$ in exponential form using the definition of a logarithm. If $x$ and $b$ are positive real numbers and $b \neq 1$ , then $\log_{b} (x) = y$ is equivalent to $b^{y} = x$ .

$e^{C} = | y \sec (x) |$

Step 3.6

Solve for $y$ .

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Step 3.6.1

Rewrite the equation as $| y \sec (x) | = e^{C}$ .

$| y \sec (x) | = e^{C}$

Step 3.6.2

Remove the absolute value term. This creates a $\pm$ on the right side of the equation because $| x | = \pm x$ .

$y \sec (x) = \pm e^{C}$

Step 3.6.3

Divide each term in $y \sec (x) = \pm e^{C}$ by $\sec (x)$ and simplify.

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Step 3.6.3.1

Divide each term in $y \sec (x) = \pm e^{C}$ by $\sec (x)$ .

$\frac{y \sec (x)}{\sec (x)} = \frac{\pm e^{C}}{\sec (x)}$

Step 3.6.3.2

Simplify the left side.

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Step 3.6.3.2.1

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

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Step 3.6.3.2.1.1

Cancel the common factor.

$\frac{y \sec (x)}{\sec (x)} = \frac{\pm e^{C}}{\sec (x)}$

Step 3.6.3.2.1.2

Divide $y$ by $1$ .

$y = \frac{\pm e^{C}}{\sec (x)}$

Step 3.6.3.3

Simplify the right side.

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Step 3.6.3.3.1

Separate fractions.

$y = \frac{\pm e^{C}}{1} \cdot \frac{1}{\sec (x)}$

Step 3.6.3.3.2

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

$y = \frac{\pm e^{C}}{1} \cdot \frac{1}{\frac{1}{\cos (x)}}$

Step 3.6.3.3.3

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

$y = \frac{\pm e^{C}}{1} (1 \cos (x))$

Step 3.6.3.3.4

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

$y = \frac{\pm e^{C}}{1} \cos (x)$

Step 3.6.3.3.5

Divide $\pm e^{C}$ by $1$ .

$y = (\pm e^{C}) \cos (x)$

Step 3.6.3.3.6

Reorder factors in $(\pm e^{C}) \cos (x)$ .

$y = \cos (x) (\pm e^{C})$

Step 4

Simplify the constant of integration.

$y = \cos (x) C$