Cálculo Exemplos

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

Etapa 1

Reescreva a equação diferencial como $\frac{d y}{d x} + M (x) y = Q (x)$ .

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Etapa 1.1

Divida cada termo em $\cos (x) \frac{d y}{d x} + y = \sin (x)$ por $\cos (x)$ .

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

Etapa 1.2

Cancele o fator comum de $\cos (x)$ .

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Etapa 1.2.1

Cancele o fator comum.

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

Etapa 1.2.2

Divida $\frac{d y}{d x}$ por $1$ .

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

Etapa 1.3

Fatore $y$ de $\frac{y}{\cos (x)}$ .

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

Etapa 1.4

Reordene $y$ e $\frac{1}{\cos (x)}$ .

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

Etapa 2

O fator de integração é definido pela fórmula $e^{\int P (x) d x}$ , em que $P (x) = \frac{1}{\cos (x)}$ .

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Etapa 2.1

Determine a integração.

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

Etapa 2.2

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

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Etapa 2.2.1

Converta de $\frac{1}{\cos (x)}$ em $\sec (x)$ .

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

Etapa 2.2.2

A integral de $\sec (x)$ com relação a $x$ é $\ln (| \sec (x) + \tan (x) |)$ .

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

Etapa 2.3

Remova a constante de integração.

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

Etapa 2.4

Potenciação e logaritmo são funções inversas.

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

Etapa 3

Multiplique cada termo pelo fator de integração $\sec (x) + \tan (x)$ .

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Etapa 3.1

Multiplique cada termo por $\sec (x) + \tan (x)$ .

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

Etapa 3.2

Simplifique cada termo.

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Etapa 3.2.1

Simplifique cada termo.

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Etapa 3.2.1.1

Reescreva $\sec (x)$ em termos de senos e cossenos.

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

Etapa 3.2.1.2

Reescreva $\tan (x)$ em termos de senos e cossenos.

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

Etapa 3.2.2

Aplique a propriedade distributiva.

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

Etapa 3.2.3

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

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

Etapa 3.2.4

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

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

Etapa 3.2.5

Simplifique cada termo.

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Etapa 3.2.5.1

Reescreva $\sec (x)$ em termos de senos e cossenos.

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

Etapa 3.2.5.2

Reescreva $\tan (x)$ em termos de senos e cossenos.

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

Etapa 3.2.6

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

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

Etapa 3.2.7

Aplique a propriedade distributiva.

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

Etapa 3.2.8

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

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Etapa 3.2.8.1

Multiplique $\frac{1}{\cos (x)}$ por $\frac{y}{\cos (x)}$ .

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

Etapa 3.2.8.2

Eleve $\cos (x)$ à potência de $1$ .

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

Etapa 3.2.8.3

Eleve $\cos (x)$ à potência de $1$ .

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

Etapa 3.2.8.4

Use a regra da multiplicação de potências $a^{m} a^{n} = a^{m + n}$ para combinar expoentes.

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

Etapa 3.2.8.5

Some $1$ e $1$ .

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

Etapa 3.2.9

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

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Etapa 3.2.9.1

Multiplique $\frac{\sin (x)}{\cos (x)}$ por $\frac{y}{\cos (x)}$ .

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

Etapa 3.2.9.2

Eleve $\cos (x)$ à potência de $1$ .

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

Etapa 3.2.9.3

Eleve $\cos (x)$ à potência de $1$ .

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

Etapa 3.2.9.4

Use a regra da multiplicação de potências $a^{m} a^{n} = a^{m + n}$ para combinar expoentes.

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

Etapa 3.2.9.5

Some $1$ e $1$ .

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

Etapa 3.3

Simplifique cada termo.

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Etapa 3.3.1

Reescreva $\sec (x)$ em termos de senos e cossenos.

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

Etapa 3.3.2

Reescreva $\tan (x)$ em termos de senos e cossenos.

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

Etapa 3.4

Aplique a propriedade distributiva.

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

Etapa 3.5

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

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Etapa 3.5.1

Multiplique $\frac{1}{\cos (x)}$ por $\frac{\sin (x)}{\cos (x)}$ .

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

Etapa 3.5.2

Eleve $\cos (x)$ à potência de $1$ .

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

Etapa 3.5.3

Eleve $\cos (x)$ à potência de $1$ .

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

Etapa 3.5.4

Use a regra da multiplicação de potências $a^{m} a^{n} = a^{m + n}$ para combinar expoentes.

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

Etapa 3.5.5

Some $1$ e $1$ .

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

Etapa 3.6

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

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Etapa 3.6.1

Multiplique $\frac{\sin (x)}{\cos (x)}$ por $\frac{\sin (x)}{\cos (x)}$ .

Etapa 3.6.2

Eleve $\sin (x)$ à potência de $1$ .

Etapa 3.6.3

Eleve $\sin (x)$ à potência de $1$ .

Etapa 3.6.4

Use a regra da multiplicação de potências $a^{m} a^{n} = a^{m + n}$ para combinar expoentes.

Etapa 3.6.5

Some $1$ e $1$ .

Etapa 3.6.6

Eleve $\cos (x)$ à potência de $1$ .

Etapa 3.6.7

Eleve $\cos (x)$ à potência de $1$ .

Etapa 3.6.8

Use a regra da multiplicação de potências $a^{m} a^{n} = a^{m + n}$ para combinar expoentes.

Etapa 3.6.9

Some $1$ e $1$ .

Etapa 3.7

Simplifique cada termo.

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Etapa 3.7.1

Separe as frações.

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

Etapa 3.7.2

Converta de $\frac{1}{\cos (x)}$ em $\sec (x)$ .

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

Etapa 3.7.3

Divida $\frac{d y}{d x}$ por $1$ .

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

Etapa 3.7.4

Separe as frações.

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

Etapa 3.7.5

Converta de $\frac{\sin (x)}{\cos (x)}$ em $\tan (x)$ .

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

Etapa 3.7.6

Divida $\frac{d y}{d x}$ por $1$ .

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

Etapa 3.7.7

Multiplique por $1$ .

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

Etapa 3.7.8

Separe as frações.

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

Etapa 3.7.9

Converta de $\frac{1}{\cos^{2} (x)}$ em $\sec^{2} (x)$ .

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

Etapa 3.7.10

Divida $y$ por $1$ .

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

Etapa 3.7.11

Fatore $\cos (x)$ de $\cos^{2} (x)$ .

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

Etapa 3.7.12

Separe as frações.

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

Etapa 3.7.13

Converta de $\frac{\sin (x)}{\cos (x)}$ em $\tan (x)$ .

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

Etapa 3.7.14

Separe as frações.

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

Etapa 3.7.15

Converta de $\frac{1}{\cos (x)}$ em $\sec (x)$ .

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

Etapa 3.7.16

Divida $y$ por $1$ .

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

Etapa 3.8

Simplifique cada termo.

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Etapa 3.8.1

Fatore $\cos (x)$ de $\cos^{2} (x)$ .

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

Etapa 3.8.2

Separe as frações.

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

Etapa 3.8.3

Converta de $\frac{\sin (x)}{\cos (x)}$ em $\tan (x)$ .

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

Etapa 3.8.4

Converta de $\frac{1}{\cos (x)}$ em $\sec (x)$ .

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

Etapa 3.8.5

Converta de $\frac{\sin^{2} (x)}{\cos^{2} (x)}$ em $\tan^{2} (x)$ .

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

Etapa 3.9

Reordene os fatores em $\frac{d y}{d x} \sec (x) + \frac{d y}{d x} \tan (x) + y \sec^{2} (x) + y \sec (x) \tan (x) = \sec (x) \tan (x) + \tan^{2} (x)$ .

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

Etapa 4

Reescreva o lado esquerdo como resultado da diferenciação de um produto.

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

Etapa 5

Determine uma integral de cada lado.

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

Etapa 6

Integre o lado esquerdo.

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

Etapa 7

Integre o lado direito.

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Etapa 7.1

Divida a integral única em várias integrais.

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

Etapa 7.2

Como a derivada de $\sec (x)$ é $\sec (x) \tan (x)$ , a integral de $\sec (x) \tan (x)$ é $\sec (x)$ .

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

Etapa 7.3

Usando a fórmula de Pitágoras, reescreva $\tan^{2} (x)$ como $- 1 + \sec^{2} (x)$ .

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

Etapa 7.4

Divida a integral única em várias integrais.

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

Etapa 7.5

Aplique a regra da constante.

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

Etapa 7.6

Como a derivada de $\tan (x)$ é $\sec^{2} (x)$ , a integral de $\sec^{2} (x)$ é $\tan (x)$ .

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

Etapa 7.7

Simplifique.

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

Etapa 8

Divida cada termo em $(\sec (x) + \tan (x)) y = \sec (x) - x + \tan (x) + C$ por $\sec (x) + \tan (x)$ e simplifique.

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Etapa 8.1

Divida cada termo em $(\sec (x) + \tan (x)) y = \sec (x) - x + \tan (x) + C$ por $\sec (x) + \tan (x)$ .

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

Etapa 8.2

Simplifique o lado esquerdo.

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Etapa 8.2.1

Cancele o fator comum de $\sec (x) + \tan (x)$ .

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Etapa 8.2.1.1

Cancele o fator comum.

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

Etapa 8.2.1.2

Divida $y$ por $1$ .

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

Etapa 8.3

Simplifique o lado direito.

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Etapa 8.3.1

Mova o número negativo para a frente da fração.

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

Etapa 8.3.2

Combine os numeradores em relação ao denominador comum.

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

Etapa 8.3.3

Combine os numeradores em relação ao denominador comum.

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

Etapa 8.3.4

Combine os numeradores em relação ao denominador comum.

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