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Integrate the rational functions(5x)/((...

Integrate the rational functions`(5x)/((x+1)(x^2-4)`

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Let `(5x​)/((x+1)(x^2+4))=(5x​)/((x+1)(x+2)(x-2))`
Let` (5x​)/((x+1)(x+2)(x-2))=A/(x+1)+B/(x+2)+C/(x−2)`
⇒`5x=A(x+2)(x−2)+B(x+1)(x−2)+C(x+1)(x+2) ......... (1)`
Substituting x=1,2, and −2 respectively in equation (1), we obtain `A=5/3​,B=−5/2​, and C=5/6​`
∴`(5x)​/((x+1)(x+2)(x-2))=5/(3*(x+1))−5/((x+2)*2)+5/((x−2)*6)​`
⇒`∫(5x)​/((x+1)(x+2)(x-2))=∫5/(3*(x+1))−∫5/((x+2)*2)+∫5/((x−2)*6)​`
=`5/(3)log(x+1)−5/(2)log(x+2)+5/(6​)*log(x−2)+C`
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