Question

integrate x/(x-1)^2(x+2) dx​

Answer 1

$\begin{gathered}\Large{\bold{\pink{\underline{Formula \: Used \::}}}} \end{gathered}$

$1. \: \boxed{ \blue {\rm \: \int \: \dfrac{1}{x} dx \: = log \: x \: + \: c}}$

$2. \: \boxed{ \blue {\rm \: \int \: {x}^{n} dx \: = \dfrac{ {x}^{n \: + \: 1} }{n + 1} + c}}$

$3. \: \boxed{ \blue {\rm \: \int \: {x}^{ - 2} dx \: = - \dfrac{ 1}{x} + c}}$

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$\large\underline\blue{\bold{ Question   }}$

$:\implies \tt \: \int \: \dfrac{x}{ {(x - 1)}^{2} (x + 2)}$

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$\large\underline\purple{\bold{Solution :- }}$

Using Partial Fraction, we get

$\tt \: Let \: \dfrac{x}{ {(x - 1)}^{2} (x + 2)} = \dfrac{a}{x - 1} + \dfrac{b}{ {(x - 1)}^{2} } + \dfrac{c}{x + 2}$

On taking LCM, we get

$\tt \:  x = a(x - 1)(x + 2) + b(x + 2) + c(x - 1)^{2}$

On substituting x = 1, we get

$:\implies \tt \: 1 = b(1 + 2)$

$:\implies \tt \boxed{ \pink{\: b = \dfrac{1}{3} }}$

On substituting x = - 2, we get

$:\implies \tt \: - \: 2 \: = c {( - 2 - 1)}^{2}$

$:\implies \tt \: - 2 = 9c$

$:\implies \tt \boxed{ \pink{ \: c \: = - \: \dfrac{2}{9} }}$

On substituting x = 0, we get

$:\implies \tt \: 0 = a( - 1)(2) + b(2) + c {( - 1)}^{2}$

$:\implies \tt \: 2a = \dfrac{2}{3} - \dfrac{2}{9}$

$:\implies \tt \: 2a \: = \: \dfrac{6 - 2}{9}$

$:\implies \tt \: 2a \: = \: \dfrac{4}{9}$

$:\implies \boxed{ \pink{\tt \: a \: = \: \dfrac{2}{9} }}$

So, given rational function can be rewritten as

$\sf \: \dfrac{x}{ {(x - 1)}^{2} (x + 2)} = \dfrac{2}{\dfrac{9}{(x - 1) } } + \dfrac{1}{\dfrac{3}{ {(x - 1)}^{2} } } - \dfrac{2}{\dfrac{9}{x + 2} }$

On integrating both sides w. r. t. x, we get

$:\implies \tt \: \int \: \dfrac{x}{ {(x - 1)}^{2} (x + 2)}$

$\tt \:  = \dfrac{2}{9} \int \: \dfrac{dx}{x - 1} + \dfrac{1}{3} \int \: \dfrac{dx}{ {(x - 1)}^{2} } - \dfrac{2}{9} \int \: \dfrac{dx}{x + 2}$

$\tt \:  = \dfrac{2}{9} log(x - 1) - \dfrac{1}{3(x - 1)} - \dfrac{2}{9} log(x + 2) + c$

$\tt \:  = \dfrac{2}{9} ( log(x - 1) - log(x + 2) - \dfrac{1}{3(x - 1)} + c$

$\bf \:  = \dfrac{2}{9} log \bigg|\dfrac{x - 1}{x + 2} \bigg | - \dfrac{1}{3(x - 1)} + c$