Calcule as integrais seguir, por substituição: (g) $\int_{e}^{e^{4}} \frac{1}{x \sqrt{\ln (x)}} d x$ (Resposta: 2 )

Passo 1

Beleza, vamos lá! A gente precisa calcular a integral definida de de até . Vamos usar o método de substituição para facilitar essa integral. Para isso, a gente vai precisar entender como trocar de variável e ajustar os limites de integração.

Passo 2

Primeiro, vamos fazer uma substituição para simplificar a expressão. Vamos deixar . A função logarítmica está dentro da raiz, então isso deve ajudar a simplificar a integral.

Passo 3

Agora, vamos derivar em relação a para encontrar . Sabemos que , então . Observe que isso aparece exatamente no numerador da nossa integral, o que é ótimo para a substituição.

Passo 4

Vamos também mudar os limites de integração para a nova variável . Quando , temos . Quando , temos . Então agora nossa integral vai de até .

Passo 5

Substituindo tudo na integral, temos:
Essa expressão é bem mais simples para integrar!

Passo 6

Agora, vamos calcular a integral de em relação a . Sabemos que . Aplicando isso aos limites de 1 a 4:

Resposta

2

Exercícios relacionados

Calcule as integrais seguir, por substituição:
(f) (Resposta:ln
Calcule as integrais seguir, por substituição:
(e) Resposta:
Calcule as integrais seguir, por substituição:
(h) (Resposta:
<html><head></head><body><div>Encontre a equação da reta que é tangente ao gráfico de f no ponto <span class="math-inline ">
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<div><span class="math-inline ">
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