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Find int(3x-2)/((x+1)^2(x+3)dx...

Find `int(3x-2)/((x+1)^2(x+3)dx`

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`(3x-2)/((x+1)^(2)(x+3))=(A)/(x+1)+(B)/((x+1)^(2))+(C )/((x+3))`
So that, `3x-2=A(x+1)(x+3)+B(x+3)+C(x+1)^(2)`
`A(x^(2)+4x+3)+B(x+3)+C(x^(2)+2x+1)`
Comparing coefficient of `x^(2)`, `x`, and constant term on both sides, we get
`{:(A+C=0),(4A+B+2C=3),(3A+3B+C=-2):}}{:(A=(11)/(4)),(impliesB=(-5)/(2)),(C=(-11)/(4)):}`
Hence, `(3x-2)/((x+1)^(2)(x+3))=(11)/(4(x+1))-(5)/(2(x+1)^(2))-(11)/(4(x+3))`
`int(3x-2)/((x+1)^(2)(x+3))dx=int(11)/(4(x+1))dx-int(5)/(2(x+1)^(2))dx-int(11)/(4(x+3))dx`
`=(11)/(4)log|(x+1)/(x+3)|+(5)/(2){(1)/(x+1)}+C`
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