Question

[Cálculo II]
Como resolver essas integrais da maneira mais simples? (Sem ser por partes)

S dx/((x-1)²(x-2))

Answer 1

 Boa tarde, Dario!

$\frac{1}{(x-1)^2(x-2)}=\frac{A}{(x-1)^2}+\frac{B}{x-1}+\frac{C}{x-2}\\\\\\\frac{1}{(x-1)^2(x-2)}=\frac{A(x-2)+B(x-1)(x-2)+C(x-1)^2}{(x-1)^2(x-2)}\\\\\\\frac{1}{(x-1)^2(x-2)}=\frac{A(x-2)+B(x^2-3x+2)+c(x^2-2x+1)}{(x-1)^2(x-2)}\\\\\\\frac{1}{(x-1)^2(x-2)}=\frac{(B+C)x^2+(A-3B-2C)x+(-2A+2B+C)}{(x-1)^2(x-2)}\\\\\\\begin{cases}B+C=0\\A-3B-2C=0\\-2A+2B+C=1\end{cases}\Rightarrow\begin{cases}\boxed{A=-1}\\\boxed{B=-1}\\\boxed{C=1}\end{cases}$

 Por conseguinte,

$\int\frac{dx}{(x-1)^2(x-2)}=\int\frac{-1}{(x-1)^2}dx+\int\frac{-1}{(x-1)}dx+\int\frac{1}{(x-2)}dx$

 (...)

Answer 2

Resposta:

Respostas: 7. f(x) = 1

5

(x

2 + 3)5 + C, 11. f(x) = 2√

1 + x + 2x

2 + C, 19. f(x) = −

1

π

cos π t + C,

2

Explicação passo-a-passo: