Question

integrated it fastr ​

Answer 1

Given Integrand:-

$\displaystyle \int \sf \dfrac{x}{(x - 1)(x - 2)(x - 3)} dx$

We know that:-

$\boxed{ \boxed{ \displaystyle \int \sf \dfrac{ {ax}^{2} + bx + c}{(x - p)(x - q)(x - r)} dx = \int \bigg( \dfrac{A}{x - p} + \dfrac{B}{x - q} + \dfrac{C}{x - r} \bigg)dx }}$

Therefore,

$\implies \displaystyle \sf \dfrac{x}{(x - 1)(x - 2)(x - 3)} = \dfrac{a}{x - 1} + \dfrac{b}{x - 2} + \dfrac{c}{x - 3} \\ \\ \implies \sf \: \dfrac{x}{ \cancel{(x - 1)(x - 2)(x - 3)}} = \dfrac{a(x - 2)(x - 3) + b(x - 1)(x - 3) + c(x - 2)(x - 1)}{ \cancel{(x - 1)(x - 2)(x -3)}} \\ \\ \implies \sf \: x = a(x - 2)(x - 3) + b(x - 1)(x - 3) + c(x - 2)(x - 1) - - - - (1)$

Putting x = 1 in equation (1),

$\longrightarrow \sf \: 1 = a(1 - 2)(1 - 3) + b(1 - 1)(1 - 3) + c(1 - 2)(1- 1) \\ \\ \longrightarrow \sf \: 1 = 2a \\ \\ \longrightarrow \sf \: a = \dfrac{1}{2}$

Putting x = 2,

$\longrightarrow \sf \: 2= a(2 - 2)(2 - 3) + b(2 - 1)(2 - 3) + c(2 - 2)(2- 1) \\ \\ \longrightarrow \sf \: 2 = - b \\ \\ \longrightarrow \sf \: b = - 2$

Putting x = 3,

$\longrightarrow \sf \: 3= a(3- 2)(3 - 3) + b(3 - 1)(3 - 3) + c(3 - 2)(3- 1) \\ \\ \longrightarrow \sf \: 3 = 2c \\ \\ \longrightarrow \sf \:c = \dfrac{3}{2}$

The integrand can be rewritten as :

$\implies \displaystyle \int \sf \dfrac{x}{(x - 1)(x - 2)(x - 3)}dx = \int \bigg( \dfrac{1}{2(x - 1)} - \dfrac{2}{x - 2} + \dfrac{3}{2(x - 3)} \bigg)dx$

We know that,

$\star \: \boxed{ \displaystyle \int \sf \dfrac{dx}{x} = ln(x) + c}$

Thus,

$\implies \boxed{ \boxed{\sf \: \dfrac{1}{2} ln(x - 1) - 2 ln(x - 2) + \dfrac{3}{2} ln(x - 3) + c}}$

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