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Evaluate: int(dx)/((x^(2)+3x+2)(sqrt(x+4...

Evaluate: `int(dx)/((x^(2)+3x+2)(sqrt(x+4))`

Text Solution

Verified by Experts

Let I `=int(dx)/((x^(2)+3x+2)(sqrt(x+4)))`
Putting `x+4=t^(2)`, and `dx=2t`, we get `I=int(2tdt)/((d^(2)-4)^(2)+3(t^(2)-4)+2)sqrt(t^(2))`
`rArr 2int(dt)/(t^(2)-5t^(2)+6)dt = 2int(dt)/(t^(4)-5t^(2)+6) dt =2int(dt)/(t^(4)-5t^(2)+6)dt =2int(dt)/((t^(2)2)(t^(3)-3)) dt =2int1/(t^(2)-1-1/(t^(2)-2)]dt`
`=1/sqrt(3)ln|(t-sqrt(3))/(t+sqrt(2))|+C` where `t=sqrt(x+4)`
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