Calculus Examples

$\cos (x) \frac{d y}{d x} + y + 3 = 0$

Step 1

Separate the variables.

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Step 1.1

Solve for $\frac{d y}{d x}$ .

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Step 1.1.1

Simplify the left side.

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Step 1.1.1.1

Reorder factors in $\cos (x) \frac{d y}{d x} + y + 3$ .

$\frac{d y}{d x} \cos (x) + y + 3 = 0$

Step 1.1.2

Move all terms not containing $\frac{d y}{d x}$ to the right side of the equation.

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Step 1.1.2.1

Subtract $y$ from both sides of the equation.

$\frac{d y}{d x} \cos (x) + 3 = - y$

Step 1.1.2.2

Subtract $3$ from both sides of the equation.

$\frac{d y}{d x} \cos (x) = - y - 3$

Step 1.1.3

Divide each term in $\frac{d y}{d x} \cos (x) = - y - 3$ by $\cos (x)$ and simplify.

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Step 1.1.3.1

Divide each term in $\frac{d y}{d x} \cos (x) = - y - 3$ by $\cos (x)$ .

$\frac{\frac{d y}{d x} \cos (x)}{\cos (x)} = \frac{- y}{\cos (x)} + \frac{- 3}{\cos (x)}$

Step 1.1.3.2

Simplify the left side.

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Step 1.1.3.2.1

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

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Step 1.1.3.2.1.1

Cancel the common factor.

$\frac{\frac{d y}{d x} \cos (x)}{\cos (x)} = \frac{- y}{\cos (x)} + \frac{- 3}{\cos (x)}$

Step 1.1.3.2.1.2

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

$\frac{d y}{d x} = \frac{- y}{\cos (x)} + \frac{- 3}{\cos (x)}$

Step 1.1.3.3

Simplify the right side.

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Step 1.1.3.3.1

Simplify each term.

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Step 1.1.3.3.1.1

Move the negative in front of the fraction.

$\frac{d y}{d x} = - \frac{y}{\cos (x)} + \frac{- 3}{\cos (x)}$

Step 1.1.3.3.1.2

Separate fractions.

$\frac{d y}{d x} = - (\frac{y}{1} \cdot \frac{1}{\cos (x)}) + \frac{- 3}{\cos (x)}$

Step 1.1.3.3.1.3

Convert from $\frac{1}{\cos (x)}$ to $\sec (x)$ .

$\frac{d y}{d x} = - (\frac{y}{1} \sec (x)) + \frac{- 3}{\cos (x)}$

Step 1.1.3.3.1.4

Divide $y$ by $1$ .

$\frac{d y}{d x} = - (y \sec (x)) + \frac{- 3}{\cos (x)}$

Step 1.1.3.3.1.5

Separate fractions.

$\frac{d y}{d x} = - y \sec (x) + \frac{- 3}{1} \cdot \frac{1}{\cos (x)}$

Step 1.1.3.3.1.6

Convert from $\frac{1}{\cos (x)}$ to $\sec (x)$ .

$\frac{d y}{d x} = - y \sec (x) + \frac{- 3}{1} \sec (x)$

Step 1.1.3.3.1.7

Divide $- 3$ by $1$ .

$\frac{d y}{d x} = - y \sec (x) - 3 \sec (x)$

Step 1.2

Factor $\sec (x)$ out of $- y \sec (x) - 3 \sec (x)$ .

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Step 1.2.1

Factor $\sec (x)$ out of $- y \sec (x)$ .

$\frac{d y}{d x} = \sec (x) (- y) - 3 \sec (x)$

Step 1.2.2

Factor $\sec (x)$ out of $- 3 \sec (x)$ .

$\frac{d y}{d x} = \sec (x) (- y) + \sec (x) \cdot - 3$

Step 1.2.3

Factor $\sec (x)$ out of $\sec (x) (- y) + \sec (x) \cdot - 3$ .

$\frac{d y}{d x} = \sec (x) (- y - 3)$

Step 1.3

Multiply both sides by $\frac{1}{- y - 3}$ .

$\frac{1}{- y - 3} \frac{d y}{d x} = \frac{1}{- y - 3} (\sec (x) (- y - 3))$

Step 1.4

Cancel the common factor of $- y - 3$ .

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Step 1.4.1

Factor $- y - 3$ out of $\sec (x) (- y - 3)$ .

$\frac{1}{- y - 3} \frac{d y}{d x} = \frac{1}{- y - 3} ((- y - 3) \sec (x))$

Step 1.4.2

Cancel the common factor.

$\frac{1}{- y - 3} \frac{d y}{d x} = \frac{1}{- y - 3} ((- y - 3) \sec (x))$

Step 1.4.3

Rewrite the expression.

$\frac{1}{- y - 3} \frac{d y}{d x} = \sec (x)$

Step 1.5

Rewrite the equation.

$\frac{1}{- y - 3} d y = \sec (x) d x$

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

$\int \frac{1}{- y - 3} d y = \int \sec (x) d x$

Step 2.2

Integrate the left side.

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Step 2.2.1

Let $u = - y - 3$ . Then $d u = - d y$ , so $- d u = d y$ . Rewrite using $u$ and $d$ $u$ .

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Step 2.2.1.1

Let $u = - y - 3$ . Find $\frac{d u}{d y}$ .

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Step 2.2.1.1.1

Rewrite.

$\frac{- 1}{1}$

Step 2.2.1.1.2

Divide $- 1$ by $1$ .

$- 1$

Step 2.2.1.2

Rewrite the problem using $u$ and $d u$ .

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

Step 2.2.2

Split the fraction into multiple fractions.

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

Step 2.2.3

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

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

Step 2.2.4

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

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

Step 2.2.5

Simplify.

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

Step 2.2.6

Replace all occurrences of $u$ with $- y - 3$ .

$- \ln (| - y - 3 |) + C_{1} = \int \sec (x) d x$

Step 2.3

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

$- \ln (| - y - 3 |) + C_{1} = \ln (| \sec (x) + \tan (x) |) + C_{2}$

Step 2.4

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

$- \ln (| - y - 3 |) = \ln (| \sec (x) + \tan (x) |) + C$

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.

$- \ln (| - y - 3 |) - \ln (| \sec (x) + \tan (x) |) = C$

Step 3.2

Add $\ln (| \sec (x) + \tan (x) |)$ to both sides of the equation.

$- \ln (| - y - 3 |) = C + \ln (| \sec (x) + \tan (x) |)$

Step 3.3

Divide each term in $- \ln (| - y - 3 |) = C + \ln (| \sec (x) + \tan (x) |)$ by $- 1$ and simplify.

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Step 3.3.1

Divide each term in $- \ln (| - y - 3 |) = C + \ln (| \sec (x) + \tan (x) |)$ by $- 1$ .

$\frac{- \ln (| - y - 3 |)}{- 1} = \frac{C}{- 1} + \frac{\ln (| \sec (x) + \tan (x) |)}{- 1}$

Step 3.3.2

Simplify the left side.

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Step 3.3.2.1

Dividing two negative values results in a positive value.

$\frac{\ln (| - y - 3 |)}{1} = \frac{C}{- 1} + \frac{\ln (| \sec (x) + \tan (x) |)}{- 1}$

Step 3.3.2.2

Divide $\ln (| - y - 3 |)$ by $1$ .

$\ln (| - y - 3 |) = \frac{C}{- 1} + \frac{\ln (| \sec (x) + \tan (x) |)}{- 1}$

Step 3.3.3

Simplify the right side.

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Step 3.3.3.1

Simplify each term.

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Step 3.3.3.1.1

Move the negative one from the denominator of $\frac{C}{- 1}$ .

$\ln (| - y - 3 |) = - 1 \cdot C + \frac{\ln (| \sec (x) + \tan (x) |)}{- 1}$

Step 3.3.3.1.2

Rewrite $- 1 \cdot C$ as $- C$ .

$\ln (| - y - 3 |) = - C + \frac{\ln (| \sec (x) + \tan (x) |)}{- 1}$

Step 3.3.3.1.3

Move the negative one from the denominator of $\frac{\ln (| \sec (x) + \tan (x) |)}{- 1}$ .

$\ln (| - y - 3 |) = - C - 1 \cdot \ln (| \sec (x) + \tan (x) |)$

Step 3.3.3.1.4

Rewrite $- 1 \cdot \ln (| \sec (x) + \tan (x) |)$ as $- \ln (| \sec (x) + \tan (x) |)$ .

$\ln (| - y - 3 |) = - C - \ln (| \sec (x) + \tan (x) |)$

Step 3.4

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

$\ln (| - y - 3 |) + \ln (| \sec (x) + \tan (x) |) = - C$

Step 3.5

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

$\ln (| - y - 3 | | \sec (x) + \tan (x) |) = - C$

Step 3.6

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

$\ln (| (- y - 3) (\sec (x) + \tan (x)) |) = - C$

Step 3.7

Expand $(- y - 3) (\sec (x) + \tan (x))$ using the FOIL Method.

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Step 3.7.1

Apply the distributive property.

$\ln (| - y (\sec (x) + \tan (x)) - 3 (\sec (x) + \tan (x)) |) = - C$

Step 3.7.2

Apply the distributive property.

$\ln (| - y \sec (x) - y \tan (x) - 3 (\sec (x) + \tan (x)) |) = - C$

Step 3.7.3

Apply the distributive property.

$\ln (| - y \sec (x) - y \tan (x) - 3 \sec (x) - 3 \tan (x) |) = - C$

Step 3.8

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

$e^{\ln (| - y \sec (x) - y \tan (x) - 3 \sec (x) - 3 \tan (x) |)} = e^{- C}$

Step 3.9

Rewrite $\ln (| - y \sec (x) - y \tan (x) - 3 \sec (x) - 3 \tan (x) |) = - C$ 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 $b^{y} = x$ .

$e^{- C} = | - y \sec (x) - y \tan (x) - 3 \sec (x) - 3 \tan (x) |$

Step 3.10

Solve for $y$ .

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Step 3.10.1

Rewrite the equation as $| - y \sec (x) - y \tan (x) - 3 \sec (x) - 3 \tan (x) | = e^{- C}$ .

$| - y \sec (x) - y \tan (x) - 3 \sec (x) - 3 \tan (x) | = e^{- C}$

Step 3.10.2

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

$- y \sec (x) - y \tan (x) - 3 \sec (x) - 3 \tan (x) = \pm e^{- C}$

Step 3.10.3

Move all terms not containing $y$ to the right side of the equation.

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Step 3.10.3.1

Add $3 \sec (x)$ to both sides of the equation.

$- y \sec (x) - y \tan (x) - 3 \tan (x) = \pm e^{- C} + 3 \sec (x)$

Step 3.10.3.2

Add $3 \tan (x)$ to both sides of the equation.

$- y \sec (x) - y \tan (x) = \pm e^{- C} + 3 \sec (x) + 3 \tan (x)$

Step 3.10.4

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

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Step 3.10.4.1

Factor $y$ out of $- y \sec (x)$ .

$y (- 1 \sec (x)) - y \tan (x) = \pm e^{- C} + 3 \sec (x) + 3 \tan (x)$

Step 3.10.4.2

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

$y (- 1 \sec (x)) + y (- 1 \tan (x)) = \pm e^{- C} + 3 \sec (x) + 3 \tan (x)$

Step 3.10.4.3

Factor $y$ out of $y (- 1 \sec (x)) + y (- 1 \tan (x))$ .

$y (- 1 \sec (x) - 1 \tan (x)) = \pm e^{- C} + 3 \sec (x) + 3 \tan (x)$

Step 3.10.5

Rewrite $- 1 \sec (x)$ as $- \sec (x)$ .

$y (- \sec (x) - 1 \tan (x)) = \pm e^{- C} + 3 \sec (x) + 3 \tan (x)$

Step 3.10.6

Rewrite $- 1 \tan (x)$ as $- \tan (x)$ .

$y (- \sec (x) - \tan (x)) = \pm e^{- C} + 3 \sec (x) + 3 \tan (x)$

Step 3.10.7

Divide each term in $y (- \sec (x) - \tan (x)) = \pm e^{- C} + 3 \sec (x) + 3 \tan (x)$ by $- \sec (x) - \tan (x)$ and simplify.

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Step 3.10.7.1

Divide each term in $y (- \sec (x) - \tan (x)) = \pm e^{- C} + 3 \sec (x) + 3 \tan (x)$ by $- \sec (x) - \tan (x)$ .

$\frac{y (- \sec (x) - \tan (x))}{- \sec (x) - \tan (x)} = \frac{\pm e^{- C}}{- \sec (x) - \tan (x)} + \frac{3 \sec (x)}{- \sec (x) - \tan (x)} + \frac{3 \tan (x)}{- \sec (x) - \tan (x)}$

Step 3.10.7.2

Simplify the left side.

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Step 3.10.7.2.1

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

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Step 3.10.7.2.1.1

Cancel the common factor.

$\frac{y (- \sec (x) - \tan (x))}{- \sec (x) - \tan (x)} = \frac{\pm e^{- C}}{- \sec (x) - \tan (x)} + \frac{3 \sec (x)}{- \sec (x) - \tan (x)} + \frac{3 \tan (x)}{- \sec (x) - \tan (x)}$

Step 3.10.7.2.1.2

Divide $y$ by $1$ .

$y = \frac{\pm e^{- C}}{- \sec (x) - \tan (x)} + \frac{3 \sec (x)}{- \sec (x) - \tan (x)} + \frac{3 \tan (x)}{- \sec (x) - \tan (x)}$

Step 3.10.7.3

Simplify the right side.

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Step 3.10.7.3.1

Simplify each term.

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Step 3.10.7.3.1.1

Factor $- 1$ out of $- \sec (x) - \tan (x)$ .

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Step 3.10.7.3.1.1.1

Factor $- 1$ out of $- \sec (x)$ .

$y = \frac{\pm e^{- C}}{- (\sec (x)) - \tan (x)} + \frac{3 \sec (x)}{- \sec (x) - \tan (x)} + \frac{3 \tan (x)}{- \sec (x) - \tan (x)}$

Step 3.10.7.3.1.1.2

Factor $- 1$ out of $- \tan (x)$ .

$y = \frac{\pm e^{- C}}{- (\sec (x)) - (\tan (x))} + \frac{3 \sec (x)}{- \sec (x) - \tan (x)} + \frac{3 \tan (x)}{- \sec (x) - \tan (x)}$

Step 3.10.7.3.1.1.3

Factor $- 1$ out of $- (\sec (x)) - (\tan (x))$ .

$y = \frac{\pm e^{- C}}{- (\sec (x) + \tan (x))} + \frac{3 \sec (x)}{- \sec (x) - \tan (x)} + \frac{3 \tan (x)}{- \sec (x) - \tan (x)}$

Step 3.10.7.3.1.2

Simplify $\frac{\pm e^{- C}}{- (\sec (x) + \tan (x))}$ .

$y = \frac{\pm e^{- C}}{\sec (x) + \tan (x)} + \frac{3 \sec (x)}{- \sec (x) - \tan (x)} + \frac{3 \tan (x)}{- \sec (x) - \tan (x)}$

Step 3.10.7.3.1.3

Factor $- 1$ out of $- \sec (x) - \tan (x)$ .

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Step 3.10.7.3.1.3.1

Factor $- 1$ out of $- \sec (x)$ .

$y = \frac{\pm e^{- C}}{\sec (x) + \tan (x)} + \frac{3 \sec (x)}{- (\sec (x)) - \tan (x)} + \frac{3 \tan (x)}{- \sec (x) - \tan (x)}$

Step 3.10.7.3.1.3.2

Factor $- 1$ out of $- \tan (x)$ .

$y = \frac{\pm e^{- C}}{\sec (x) + \tan (x)} + \frac{3 \sec (x)}{- (\sec (x)) - (\tan (x))} + \frac{3 \tan (x)}{- \sec (x) - \tan (x)}$

Step 3.10.7.3.1.3.3

Factor $- 1$ out of $- (\sec (x)) - (\tan (x))$ .

$y = \frac{\pm e^{- C}}{\sec (x) + \tan (x)} + \frac{3 \sec (x)}{- (\sec (x) + \tan (x))} + \frac{3 \tan (x)}{- \sec (x) - \tan (x)}$

Step 3.10.7.3.1.4

Move the negative in front of the fraction.

$y = \frac{\pm e^{- C}}{\sec (x) + \tan (x)} - \frac{3 \sec (x)}{\sec (x) + \tan (x)} + \frac{3 \tan (x)}{- \sec (x) - \tan (x)}$

Step 3.10.7.3.1.5

Factor $- 1$ out of $- \sec (x) - \tan (x)$ .

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Step 3.10.7.3.1.5.1

Factor $- 1$ out of $- \sec (x)$ .

$y = \frac{\pm e^{- C}}{\sec (x) + \tan (x)} - \frac{3 \sec (x)}{\sec (x) + \tan (x)} + \frac{3 \tan (x)}{- (\sec (x)) - \tan (x)}$

Step 3.10.7.3.1.5.2

Factor $- 1$ out of $- \tan (x)$ .

$y = \frac{\pm e^{- C}}{\sec (x) + \tan (x)} - \frac{3 \sec (x)}{\sec (x) + \tan (x)} + \frac{3 \tan (x)}{- (\sec (x)) - (\tan (x))}$

Step 3.10.7.3.1.5.3

Factor $- 1$ out of $- (\sec (x)) - (\tan (x))$ .

$y = \frac{\pm e^{- C}}{\sec (x) + \tan (x)} - \frac{3 \sec (x)}{\sec (x) + \tan (x)} + \frac{3 \tan (x)}{- (\sec (x) + \tan (x))}$

Step 3.10.7.3.1.6

Move the negative in front of the fraction.

$y = \frac{\pm e^{- C}}{\sec (x) + \tan (x)} - \frac{3 \sec (x)}{\sec (x) + \tan (x)} - \frac{3 \tan (x)}{\sec (x) + \tan (x)}$

Step 3.10.7.3.2

Combine into one fraction.

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Step 3.10.7.3.2.1

Combine the numerators over the common denominator.

$y = \frac{\pm e^{- C} - 3 \sec (x)}{\sec (x) + \tan (x)} - \frac{3 \tan (x)}{\sec (x) + \tan (x)}$

Step 3.10.7.3.2.2

Combine the numerators over the common denominator.

$y = \frac{\pm e^{- C} - 3 \sec (x) - 3 \tan (x)}{\sec (x) + \tan (x)}$

Step 4

Simplify the constant of integration.

$y = \frac{C - 3 \sec (x) - 3 \tan (x)}{\sec (x) + \tan (x)}$