Question

Can anyone just help me out with this. Would be thankful !

Answer 1

$\int \frac{2 {x}^{2} + 3 }{( {x}^{2} - 1)( {x}^{2} + 4) } \\ = \int \frac{1 }{( {x}^{2} - 1)} + \int \frac{1 }{( {x}^{2} + 4) } \\ = \int \frac{1 }{(x + 1)( x- 1)} + \int \frac{1 }{( {x}^{2} + {2}^{2} ) } \\ = \frac{1}{2} \int (\frac{1 }{( x- 1)} - \frac{1 }{(x + 1)} ) + \frac{1}{2} \int \frac{2 }{( {x}^{2} + {2}^{2} ) }\\ = \frac{1}{2} ( ln(x - 1) - ln((x + 1)) + \frac{1}{2} {\tan}^{ - 1} ( \frac{x}{2} ) \\ = \frac{1}{2} ln( \frac{x - 1}{x + 1} ) + \frac{1}{2} {\tan}^{ - 1} ( \frac{x}{2} ) + c \\ comparing \: \: a = \frac{1}{2} \: \: and \: \: b = \frac{1}{2}$