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Question

Integrate
dx(1+x)3+2xx2

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Solution

Consider the given integral.


I=dx(1+x)3+2xx2 ……. (1)



Let t=11+x


dtdx=1(1+x)2


dx=(1+x)2dt



Therefore,


I=(1+x)2dt(1+x)3+2xx2


I=dtt3+2(1tt)(1tt)2


I=dtt3+2(1tt)(1+t22tt2)


I=dtt(3t2+2t2t21t2+2tt2)


I=dt4t1



Let p=4t1


dpdt=4


dp4=dt



Therefore,


I=14dpp


I=14(2p)+C


I=12(p)+C



On putting the value of p, we get


I=12(4t1)+C



On putting the value of t, we get


I=12(4(11+x)1)+C


I=12(41x1+x)+C


I=12(3x1+x)+C



Hence, this is the answer.


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