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Let `f(x) = (12 + 8x-x^2) /((2-x)(4 + x^2))` Show that `int_0^1f(x)dx = ln(25/2)`

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First we need to use partial fractions to break the function in to two.





By comparing coefficients;

`x^2rarr -1=A-B` ------------(1)

`xrarr 8=2B-C` ----------------(2)

`Constant rarr 12 =2C+4A` -------(3)

solving (1),(2) and (3) give you;




`12+8x-x^2/(2-x)(4+x^2) = 3/(2-x) + 4x/(4+x^2)`

`int_0^1 f(x) dx `

`= int_0^1 3/(2-x)dx + int_0^1 (4x)/(4+x^2)dx`









So the required answer is proved.

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