Calcule a integral usando a Regra da Substituição: $[8] \int \frac{1}{(3-5 x)^{2}} d x$

Passo 1

Então, temos a integral . Vamos resolver isso usando a Regra da Substituição, que é uma técnica de cálculo para simplificar integrais. Para aplicar essa regra, precisamos identificar uma parte da integral que pode ser substituída por uma nova variável, facilitando o cálculo.

Passo 2

Primeiro, vamos escolher uma substituição apropriada. Olhando para a expressão dentro do denominador, , parece uma boa candidata. Então, vamos definir:

Passo 3

Agora, precisamos encontrar a derivada de em relação a para substituir . Derivando ambos os lados em relação a :

Passo 4

Substituímos e na integral original:

Passo 5

Agora, integramos em relação a :
Finalmente, substituímos de volta para :

Resposta

A integral é igual a , onde é a constante de integração.

Exercícios relacionados

Calcule a integral usando a Regra da Substituição:
Calcule a integral usando a Regra da Substituição:
<html><head></head><body><div>Esboce a região delimitada pelas curvas e encontre sua área:<br>
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Seja , onde e é uma função diferenciável em e tal que
Calcule quando e .