Resolve a integral abaixo:

39 –

∫ - π / 3 0 2 sec 3 ⁡ x   d x

Passo 1

2 ∫ - π / 3 0 sec 3 ⁡ x   d x

Vamos resolver a integral indefinida de:

∫ sec 3 ⁡ x   d x  

Utilizando a fórmula:

∫ sec n ⁡ x   d x = 1 n - 1 ⋅ sec n - 2 ⁡ x ⋅   t g   ( x ) + n - 2 n - 1 ∫ sec n - 2 ⁡ x   d x

Substituindo na fórmula teremos:

∫ sec 3 ⁡ x   d x = 1 3 - 1 ⋅ sec 3 - 2 ⁡ x ⋅   t g   ( x ) + 3 - 2 3 - 1 ∫ sec 3 - 2 ⁡ x   d x

Resolvendo:

1 2 ⋅ sec 1 ⁡ x ⋅   t g   ( x ) + 1 2 ∫ sec ⁡ x   d x

Passo 2

Seja:

∫ sec ⁡ x   d x = ln ⁡ sec ⁡ x + t g   ( x )

Assim:

1 2 ⋅ sec 1 ⁡ x ⋅   t g   ( x ) + 1 2 ln ⁡ sec ⁡ x + t g   ( x )

Passo 3

Resolvendo a integral com os limites de integração:

2 1 2 ⋅ sec ⁡ x ⋅   t g   x + 1 2 ln ⁡ sec ⁡ x + t g   ( x ) - π 3 0

Substituindo com os limites:

2 1 2 ⋅ sec ⁡ 0 ⋅   t g   0 + 1 2 ln ⁡ sec ⁡ 0 + t g   0 - 1 2 ⋅ sec ⁡ - π 3 ⋅   t g   - π 3 + 1 2 ln ⁡ sec ⁡ - π 3 + t g   - π 3

Seja:

sec ⁡ ( 0 ) = 1 ; sec ⁡ - π 3 = 2   ; t g   0 = 0   ; t g   - π 3 = - 3 ; ln ⁡ 1 = 0

Substituindo:

2 1 2 ⋅ 1 ⋅   0 + 1 2 ln ⁡ 1 + 0 - 1 2 ⋅ 2 ⋅   - 3 + 1 2 ln ⁡ 2 - 3

2 -   - 3 + 1 2 ln ⁡ 2 - 3 = 2 3 - 1 2   ln ⁡ 2 - 3

Multiplicando por 2:

2 3 - ln ⁡ 2 - 3

Resposta

2 3 - ln ⁡ 2 - 3

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