Calculus Examples

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

Step 1

Rewrite the differential equation as $\frac{d y}{d x} + M (x) y = Q (x)$ .

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Step 1.1

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

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

Step 1.2

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

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Step 1.2.1

Cancel the common factor.

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

Step 1.2.2

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

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

Step 1.3

Factor $y$ out of $\frac{y}{\cos (x)}$ .

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

Step 1.4

Reorder $y$ and $\frac{1}{\cos (x)}$ .

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

Step 2

The integrating factor is defined by the formula $e^{\int P (x) d x}$ , where $P (x) = \frac{1}{\cos (x)}$ .

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Step 2.1

Set up the integration.

$e^{\int \frac{1}{\cos (x)} d x}$

Step 2.2

Integrate $\frac{1}{\cos (x)}$ .

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Step 2.2.1

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

$e^{\int \sec (x) d x}$

Step 2.2.2

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

$e^{\ln (| \sec (x) + \tan (x) |) + C}$

Step 2.3

Remove the constant of integration.

$e^{\ln (\sec (x) + \tan (x))}$

Step 2.4

Exponentiation and log are inverse functions.

$\sec (x) + \tan (x)$

Step 3

Multiply each term by the integrating factor $\sec (x) + \tan (x)$ .

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Step 3.1

Multiply each term by $\sec (x) + \tan (x)$ .

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

Step 3.2

Simplify each term.

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Step 3.2.1

Simplify each term.

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Step 3.2.1.1

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

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

Step 3.2.1.2

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

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

Step 3.2.2

Apply the distributive property.

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

Step 3.2.3

Combine $\frac{1}{\cos (x)}$ and $\frac{d y}{d x}$ .

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

Step 3.2.4

Combine $\frac{\sin (x)}{\cos (x)}$ and $\frac{d y}{d x}$ .

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

Step 3.2.5

Simplify each term.

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Step 3.2.5.1

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

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

Step 3.2.5.2

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

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

Step 3.2.6

Combine $\frac{1}{\cos (x)}$ and $y$ .

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

Step 3.2.7

Apply the distributive property.

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

Step 3.2.8

Multiply $\frac{1}{\cos (x)} \cdot \frac{y}{\cos (x)}$ .

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Step 3.2.8.1

Multiply $\frac{1}{\cos (x)}$ by $\frac{y}{\cos (x)}$ .

$\frac{\frac{d y}{d x}}{\cos (x)} + \frac{\sin (x) \frac{d y}{d x}}{\cos (x)} + \frac{y}{\cos (x) \cos (x)} + \frac{\sin (x)}{\cos (x)} \cdot \frac{y}{\cos (x)} = (\sec (x) + \tan (x)) \frac{\sin (x)}{\cos (x)}$

Step 3.2.8.2

Raise $\cos (x)$ to the power of $1$ .

$\frac{\frac{d y}{d x}}{\cos (x)} + \frac{\sin (x) \frac{d y}{d x}}{\cos (x)} + \frac{y}{\cos^{1} (x) \cos (x)} + \frac{\sin (x)}{\cos (x)} \cdot \frac{y}{\cos (x)} = (\sec (x) + \tan (x)) \frac{\sin (x)}{\cos (x)}$

Step 3.2.8.3

Raise $\cos (x)$ to the power of $1$ .

$\frac{\frac{d y}{d x}}{\cos (x)} + \frac{\sin (x) \frac{d y}{d x}}{\cos (x)} + \frac{y}{\cos^{1} (x) \cos^{1} (x)} + \frac{\sin (x)}{\cos (x)} \cdot \frac{y}{\cos (x)} = (\sec (x) + \tan (x)) \frac{\sin (x)}{\cos (x)}$

Step 3.2.8.4

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{\frac{d y}{d x}}{\cos (x)} + \frac{\sin (x) \frac{d y}{d x}}{\cos (x)} + \frac{y}{{\cos (x)}^{1 + 1}} + \frac{\sin (x)}{\cos (x)} \cdot \frac{y}{\cos (x)} = (\sec (x) + \tan (x)) \frac{\sin (x)}{\cos (x)}$

Step 3.2.8.5

Add $1$ and $1$ .

$\frac{\frac{d y}{d x}}{\cos (x)} + \frac{\sin (x) \frac{d y}{d x}}{\cos (x)} + \frac{y}{\cos^{2} (x)} + \frac{\sin (x)}{\cos (x)} \cdot \frac{y}{\cos (x)} = (\sec (x) + \tan (x)) \frac{\sin (x)}{\cos (x)}$

Step 3.2.9

Multiply $\frac{\sin (x)}{\cos (x)} \cdot \frac{y}{\cos (x)}$ .

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Step 3.2.9.1

Multiply $\frac{\sin (x)}{\cos (x)}$ by $\frac{y}{\cos (x)}$ .

$\frac{\frac{d y}{d x}}{\cos (x)} + \frac{\sin (x) \frac{d y}{d x}}{\cos (x)} + \frac{y}{\cos^{2} (x)} + \frac{\sin (x) y}{\cos (x) \cos (x)} = (\sec (x) + \tan (x)) \frac{\sin (x)}{\cos (x)}$

Step 3.2.9.2

Raise $\cos (x)$ to the power of $1$ .

$\frac{\frac{d y}{d x}}{\cos (x)} + \frac{\sin (x) \frac{d y}{d x}}{\cos (x)} + \frac{y}{\cos^{2} (x)} + \frac{\sin (x) y}{\cos^{1} (x) \cos (x)} = (\sec (x) + \tan (x)) \frac{\sin (x)}{\cos (x)}$

Step 3.2.9.3

Raise $\cos (x)$ to the power of $1$ .

$\frac{\frac{d y}{d x}}{\cos (x)} + \frac{\sin (x) \frac{d y}{d x}}{\cos (x)} + \frac{y}{\cos^{2} (x)} + \frac{\sin (x) y}{\cos^{1} (x) \cos^{1} (x)} = (\sec (x) + \tan (x)) \frac{\sin (x)}{\cos (x)}$

Step 3.2.9.4

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{\frac{d y}{d x}}{\cos (x)} + \frac{\sin (x) \frac{d y}{d x}}{\cos (x)} + \frac{y}{\cos^{2} (x)} + \frac{\sin (x) y}{{\cos (x)}^{1 + 1}} = (\sec (x) + \tan (x)) \frac{\sin (x)}{\cos (x)}$

Step 3.2.9.5

Add $1$ and $1$ .

$\frac{\frac{d y}{d x}}{\cos (x)} + \frac{\sin (x) \frac{d y}{d x}}{\cos (x)} + \frac{y}{\cos^{2} (x)} + \frac{\sin (x) y}{\cos^{2} (x)} = (\sec (x) + \tan (x)) \frac{\sin (x)}{\cos (x)}$

Step 3.3

Simplify each term.

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Step 3.3.1

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

$\frac{\frac{d y}{d x}}{\cos (x)} + \frac{\sin (x) \frac{d y}{d x}}{\cos (x)} + \frac{y}{\cos^{2} (x)} + \frac{\sin (x) y}{\cos^{2} (x)} = (\frac{1}{\cos (x)} + \tan (x)) \frac{\sin (x)}{\cos (x)}$

Step 3.3.2

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

$\frac{\frac{d y}{d x}}{\cos (x)} + \frac{\sin (x) \frac{d y}{d x}}{\cos (x)} + \frac{y}{\cos^{2} (x)} + \frac{\sin (x) y}{\cos^{2} (x)} = (\frac{1}{\cos (x)} + \frac{\sin (x)}{\cos (x)}) \frac{\sin (x)}{\cos (x)}$

Step 3.4

Apply the distributive property.

$\frac{\frac{d y}{d x}}{\cos (x)} + \frac{\sin (x) \frac{d y}{d x}}{\cos (x)} + \frac{y}{\cos^{2} (x)} + \frac{\sin (x) y}{\cos^{2} (x)} = \frac{1}{\cos (x)} \cdot \frac{\sin (x)}{\cos (x)} + \frac{\sin (x)}{\cos (x)} \cdot \frac{\sin (x)}{\cos (x)}$

Step 3.5

Multiply $\frac{1}{\cos (x)} \cdot \frac{\sin (x)}{\cos (x)}$ .

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Step 3.5.1

Multiply $\frac{1}{\cos (x)}$ by $\frac{\sin (x)}{\cos (x)}$ .

$\frac{\frac{d y}{d x}}{\cos (x)} + \frac{\sin (x) \frac{d y}{d x}}{\cos (x)} + \frac{y}{\cos^{2} (x)} + \frac{\sin (x) y}{\cos^{2} (x)} = \frac{\sin (x)}{\cos (x) \cos (x)} + \frac{\sin (x)}{\cos (x)} \cdot \frac{\sin (x)}{\cos (x)}$

Step 3.5.2

Raise $\cos (x)$ to the power of $1$ .

$\frac{\frac{d y}{d x}}{\cos (x)} + \frac{\sin (x) \frac{d y}{d x}}{\cos (x)} + \frac{y}{\cos^{2} (x)} + \frac{\sin (x) y}{\cos^{2} (x)} = \frac{\sin (x)}{\cos^{1} (x) \cos (x)} + \frac{\sin (x)}{\cos (x)} \cdot \frac{\sin (x)}{\cos (x)}$

Step 3.5.3

Raise $\cos (x)$ to the power of $1$ .

$\frac{\frac{d y}{d x}}{\cos (x)} + \frac{\sin (x) \frac{d y}{d x}}{\cos (x)} + \frac{y}{\cos^{2} (x)} + \frac{\sin (x) y}{\cos^{2} (x)} = \frac{\sin (x)}{\cos^{1} (x) \cos^{1} (x)} + \frac{\sin (x)}{\cos (x)} \cdot \frac{\sin (x)}{\cos (x)}$

Step 3.5.4

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{\frac{d y}{d x}}{\cos (x)} + \frac{\sin (x) \frac{d y}{d x}}{\cos (x)} + \frac{y}{\cos^{2} (x)} + \frac{\sin (x) y}{\cos^{2} (x)} = \frac{\sin (x)}{{\cos (x)}^{1 + 1}} + \frac{\sin (x)}{\cos (x)} \cdot \frac{\sin (x)}{\cos (x)}$

Step 3.5.5

Add $1$ and $1$ .

$\frac{\frac{d y}{d x}}{\cos (x)} + \frac{\sin (x) \frac{d y}{d x}}{\cos (x)} + \frac{y}{\cos^{2} (x)} + \frac{\sin (x) y}{\cos^{2} (x)} = \frac{\sin (x)}{\cos^{2} (x)} + \frac{\sin (x)}{\cos (x)} \cdot \frac{\sin (x)}{\cos (x)}$

Step 3.6

Multiply $\frac{\sin (x)}{\cos (x)} \cdot \frac{\sin (x)}{\cos (x)}$ .

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Step 3.6.1

Multiply $\frac{\sin (x)}{\cos (x)}$ by $\frac{\sin (x)}{\cos (x)}$ .

Step 3.6.2

Raise $\sin (x)$ to the power of $1$ .

Step 3.6.3

Raise $\sin (x)$ to the power of $1$ .

Step 3.6.4

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

Step 3.6.5

Add $1$ and $1$ .

Step 3.6.6

Raise $\cos (x)$ to the power of $1$ .

Step 3.6.7

Raise $\cos (x)$ to the power of $1$ .

Step 3.6.8

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

Step 3.6.9

Add $1$ and $1$ .

Step 3.7

Simplify each term.

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Step 3.7.1

Separate fractions.

$\frac{\frac{d y}{d x}}{1} \cdot \frac{1}{\cos (x)} + \frac{\sin (x) \frac{d y}{d x}}{\cos (x)} + \frac{y}{\cos^{2} (x)} + \frac{\sin (x) y}{\cos^{2} (x)} = \frac{\sin (x)}{\cos^{2} (x)} + \frac{\sin^{2} (x)}{\cos^{2} (x)}$

Step 3.7.2

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

$\frac{\frac{d y}{d x}}{1} \sec (x) + \frac{\sin (x) \frac{d y}{d x}}{\cos (x)} + \frac{y}{\cos^{2} (x)} + \frac{\sin (x) y}{\cos^{2} (x)} = \frac{\sin (x)}{\cos^{2} (x)} + \frac{\sin^{2} (x)}{\cos^{2} (x)}$

Step 3.7.3

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

$\frac{d y}{d x} \sec (x) + \frac{\sin (x) \frac{d y}{d x}}{\cos (x)} + \frac{y}{\cos^{2} (x)} + \frac{\sin (x) y}{\cos^{2} (x)} = \frac{\sin (x)}{\cos^{2} (x)} + \frac{\sin^{2} (x)}{\cos^{2} (x)}$

Step 3.7.4

Separate fractions.

$\frac{d y}{d x} \sec (x) + \frac{\frac{d y}{d x}}{1} \cdot \frac{\sin (x)}{\cos (x)} + \frac{y}{\cos^{2} (x)} + \frac{\sin (x) y}{\cos^{2} (x)} = \frac{\sin (x)}{\cos^{2} (x)} + \frac{\sin^{2} (x)}{\cos^{2} (x)}$

Step 3.7.5

Convert from $\frac{\sin (x)}{\cos (x)}$ to $\tan (x)$ .

$\frac{d y}{d x} \sec (x) + \frac{\frac{d y}{d x}}{1} \tan (x) + \frac{y}{\cos^{2} (x)} + \frac{\sin (x) y}{\cos^{2} (x)} = \frac{\sin (x)}{\cos^{2} (x)} + \frac{\sin^{2} (x)}{\cos^{2} (x)}$

Step 3.7.6

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

$\frac{d y}{d x} \sec (x) + \frac{d y}{d x} \tan (x) + \frac{y}{\cos^{2} (x)} + \frac{\sin (x) y}{\cos^{2} (x)} = \frac{\sin (x)}{\cos^{2} (x)} + \frac{\sin^{2} (x)}{\cos^{2} (x)}$

Step 3.7.7

Multiply by $1$ .

$\frac{d y}{d x} \sec (x) + \frac{d y}{d x} \tan (x) + \frac{y}{\cos^{2} (x) \cdot 1} + \frac{\sin (x) y}{\cos^{2} (x)} = \frac{\sin (x)}{\cos^{2} (x)} + \frac{\sin^{2} (x)}{\cos^{2} (x)}$

Step 3.7.8

Separate fractions.

$\frac{d y}{d x} \sec (x) + \frac{d y}{d x} \tan (x) + \frac{y}{1} \cdot \frac{1}{\cos^{2} (x)} + \frac{\sin (x) y}{\cos^{2} (x)} = \frac{\sin (x)}{\cos^{2} (x)} + \frac{\sin^{2} (x)}{\cos^{2} (x)}$

Step 3.7.9

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

$\frac{d y}{d x} \sec (x) + \frac{d y}{d x} \tan (x) + \frac{y}{1} \sec^{2} (x) + \frac{\sin (x) y}{\cos^{2} (x)} = \frac{\sin (x)}{\cos^{2} (x)} + \frac{\sin^{2} (x)}{\cos^{2} (x)}$

Step 3.7.10

Divide $y$ by $1$ .

$\frac{d y}{d x} \sec (x) + \frac{d y}{d x} \tan (x) + y \sec^{2} (x) + \frac{\sin (x) y}{\cos^{2} (x)} = \frac{\sin (x)}{\cos^{2} (x)} + \frac{\sin^{2} (x)}{\cos^{2} (x)}$

Step 3.7.11

Factor $\cos (x)$ out of $\cos^{2} (x)$ .

$\frac{d y}{d x} \sec (x) + \frac{d y}{d x} \tan (x) + y \sec^{2} (x) + \frac{\sin (x) y}{\cos (x) \cos (x)} = \frac{\sin (x)}{\cos^{2} (x)} + \frac{\sin^{2} (x)}{\cos^{2} (x)}$

Step 3.7.12

Separate fractions.

$\frac{d y}{d x} \sec (x) + \frac{d y}{d x} \tan (x) + y \sec^{2} (x) + \frac{y}{\cos (x)} \cdot \frac{\sin (x)}{\cos (x)} = \frac{\sin (x)}{\cos^{2} (x)} + \frac{\sin^{2} (x)}{\cos^{2} (x)}$

Step 3.7.13

Convert from $\frac{\sin (x)}{\cos (x)}$ to $\tan (x)$ .

$\frac{d y}{d x} \sec (x) + \frac{d y}{d x} \tan (x) + y \sec^{2} (x) + \frac{y}{\cos (x)} \tan (x) = \frac{\sin (x)}{\cos^{2} (x)} + \frac{\sin^{2} (x)}{\cos^{2} (x)}$

Step 3.7.14

Separate fractions.

$\frac{d y}{d x} \sec (x) + \frac{d y}{d x} \tan (x) + y \sec^{2} (x) + \frac{y}{1} \cdot \frac{1}{\cos (x)} \tan (x) = \frac{\sin (x)}{\cos^{2} (x)} + \frac{\sin^{2} (x)}{\cos^{2} (x)}$

Step 3.7.15

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

$\frac{d y}{d x} \sec (x) + \frac{d y}{d x} \tan (x) + y \sec^{2} (x) + \frac{y}{1} \sec (x) \tan (x) = \frac{\sin (x)}{\cos^{2} (x)} + \frac{\sin^{2} (x)}{\cos^{2} (x)}$

Step 3.7.16

Divide $y$ by $1$ .

$\frac{d y}{d x} \sec (x) + \frac{d y}{d x} \tan (x) + y \sec^{2} (x) + y \sec (x) \tan (x) = \frac{\sin (x)}{\cos^{2} (x)} + \frac{\sin^{2} (x)}{\cos^{2} (x)}$

Step 3.8

Simplify each term.

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Step 3.8.1

Factor $\cos (x)$ out of $\cos^{2} (x)$ .

$\frac{d y}{d x} \sec (x) + \frac{d y}{d x} \tan (x) + y \sec^{2} (x) + y \sec (x) \tan (x) = \frac{\sin (x)}{\cos (x) \cos (x)} + \frac{\sin^{2} (x)}{\cos^{2} (x)}$

Step 3.8.2

Separate fractions.

$\frac{d y}{d x} \sec (x) + \frac{d y}{d x} \tan (x) + y \sec^{2} (x) + y \sec (x) \tan (x) = \frac{1}{\cos (x)} \cdot \frac{\sin (x)}{\cos (x)} + \frac{\sin^{2} (x)}{\cos^{2} (x)}$

Step 3.8.3

Convert from $\frac{\sin (x)}{\cos (x)}$ to $\tan (x)$ .

$\frac{d y}{d x} \sec (x) + \frac{d y}{d x} \tan (x) + y \sec^{2} (x) + y \sec (x) \tan (x) = \frac{1}{\cos (x)} \tan (x) + \frac{\sin^{2} (x)}{\cos^{2} (x)}$

Step 3.8.4

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

$\frac{d y}{d x} \sec (x) + \frac{d y}{d x} \tan (x) + y \sec^{2} (x) + y \sec (x) \tan (x) = \sec (x) \tan (x) + \frac{\sin^{2} (x)}{\cos^{2} (x)}$

Step 3.8.5

Convert from $\frac{\sin^{2} (x)}{\cos^{2} (x)}$ to $\tan^{2} (x)$ .

$\frac{d y}{d x} \sec (x) + \frac{d y}{d x} \tan (x) + y \sec^{2} (x) + y \sec (x) \tan (x) = \sec (x) \tan (x) + \tan^{2} (x)$

Step 3.9

Reorder factors in $\frac{d y}{d x} \sec (x) + \frac{d y}{d x} \tan (x) + y \sec^{2} (x) + y \sec (x) \tan (x) = \sec (x) \tan (x) + \tan^{2} (x)$ .

$\sec (x) \frac{d y}{d x} + \tan (x) \frac{d y}{d x} + y \sec^{2} (x) + y \sec (x) \tan (x) = \sec (x) \tan (x) + \tan^{2} (x)$

Step 4

Rewrite the left side as a result of differentiating a product.

$\frac{d}{d x} [(\sec (x) + \tan (x)) y] = \sec (x) \tan (x) + \tan^{2} (x)$

Step 5

Set up an integral on each side.

$\int \frac{d}{d x} [(\sec (x) + \tan (x)) y] d x = \int \sec (x) \tan (x) + \tan^{2} (x) d x$

Step 6

Integrate the left side.

$(\sec (x) + \tan (x)) y = \int \sec (x) \tan (x) + \tan^{2} (x) d x$

Step 7

Integrate the right side.

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Step 7.1

Split the single integral into multiple integrals.

$(\sec (x) + \tan (x)) y = \int \sec (x) \tan (x) d x + \int \tan^{2} (x) d x$

Step 7.2

Since the derivative of $\sec (x)$ is $\sec (x) \tan (x)$ , the integral of $\sec (x) \tan (x)$ is $\sec (x)$ .

$(\sec (x) + \tan (x)) y = \sec (x) + C + \int \tan^{2} (x) d x$

Step 7.3

Using the Pythagorean Identity, rewrite $\tan^{2} (x)$ as $- 1 + \sec^{2} (x)$ .

$(\sec (x) + \tan (x)) y = \sec (x) + C + \int - 1 + \sec^{2} (x) d x$

Step 7.4

Split the single integral into multiple integrals.

$(\sec (x) + \tan (x)) y = \sec (x) + C + \int - 1 d x + \int \sec^{2} (x) d x$

Step 7.5

Apply the constant rule.

$(\sec (x) + \tan (x)) y = \sec (x) + C - x + C + \int \sec^{2} (x) d x$

Step 7.6

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

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

Step 7.7

Simplify.

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

Step 8

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

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Step 8.1

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

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

Step 8.2

Simplify the left side.

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Step 8.2.1

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

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Step 8.2.1.1

Cancel the common factor.

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

Step 8.2.1.2

Divide $y$ by $1$ .

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

Step 8.3

Simplify the right side.

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Step 8.3.1

Move the negative in front of the fraction.

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

Step 8.3.2

Combine the numerators over the common denominator.

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

Step 8.3.3

Combine the numerators over the common denominator.

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

Step 8.3.4

Combine the numerators over the common denominator.

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