Question

2x+3/(2-x)(x^2+3) integration

Answer 1

$\large \green{ \rm \:SOLUTION }$

$\sf \blue{ \int \frac{2x + 3}{(2 - x)( {x}^{2} + 3)} }$

$\sf \pink{First \: convert \: it \: into \: partial \: fraction }$

$\sf \blue{ \rightarrow\frac{2x + 3}{(2 - x)( {x}^{2} + 3)} = \frac{a}{2 - x} + \frac{bx + c}{ {x}^{2} + 3 } }$

$\sf \blue{ \rightarrow 2x + 3= a( {x}^{2} + 3) + (bx + c)(2 - x) }$

$\sf \blue{ \rightarrow 2x + 3= a{x}^{2} + 3a+ 2bx + 2c - {bx}^{2} - cx }$

$\sf \blue{ \rightarrow 2x + 3= {x}^{2}(a - b) + (2b - c)x +3a + 2c }$

$\sf \pink{ Now \: comparing \: both \: sides}$

$\sf \orange {\: a - b= 0}$

$\sf \orange {\: 2b- c= 2}$

$\sf \orange {\: 3a - 2c= 3}$

$\sf \orange {\: a = b = 1}$

$\sf \orange {\: c=0}$

$\sf \blue{ \rightarrow\frac{2x + 3}{(2 - x)( {x}^{2} + 3)} = \frac{1}{2 - x} + \frac{x }{ {x}^{2} + 3 } }$

$\sf \blue{ \rightarrow \int\frac{2x + 3}{(2 - x)( {x}^{2} + 3)} = \int \frac{dx}{2 - x} + \int\frac{x dx}{ {x}^{2} + 3 } }$

$\sf \blue{ = \int \frac{dx}{2 - x} + \int\frac{x dx}{ {x}^{2} + 3 } }$

$\sf \blue{ = - log(2 - x) + \frac{1}{2} log( {x}^{2} + 3) + c }$

$\sf \green{ \int \frac{2x + 3}{(2 - x)( {x}^{2} + 3)} = - log(2 - x) + \frac{1}{2} log( {x}^{2} + 3) + c }$