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Question

The solution of dxx2+c using Euler's substitution is?

A
logxx2+c
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B
logx+x2+c
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C
logx+x2c
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D
logxx2c
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Solution

The correct option is A logx+x2+c
We have to find the solution of dxx2+c using Euler substitution.
Consider dxx2+c
We can use Euler first substitution: x2+c=x+t
x2+c=(x+t)2
x2+c=x22xt+t2
x=t2c2t
Differentiating both sides we get
dx=t2+c2t2dt
Also x2+c=t2c2t+t=t2+c2t
dxx2+c=t2+c2t2dtt2+c2t
=dtt
=log|t|
=log|x+x2+c|
Hence dxx2+c=log|x+x2+c|

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