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Question

If I=dx1+x2+2x+2=A7logx+1+x2+2x+2x2+2x+21x+1+C
then A is equal to

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Solution

I=dx1+x2+2x+2=dx(x+1)2+1+1=(x+1)2+11(x+1)2dx
Substitute (x+1)2+1=t2
I=(t1)t(t21)t21dt=(t)dt(t+1)t21=dtt21dt(t+1)t21

I=logt+t21I1
Where I1=dt(t+1)t21
Put t+1=1u
I1=(1u2)du(1u)(1u1)21=du12u=12u= t1t+1
Therefore,
I=logt+t21t1t21+c

I=logx+1+x2+2x+2x2+2x+21x+1+c

Comparing with, I=dx1+x2+2x+2=A7logx+1+x2+2x+2x2+2x+21x+1+C
Therefore, A=7

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