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Question

Evaluate: dxx2x+2 using Euler's substitution.

A
2tan11xx+2+C
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B
2tan11+xx+2+C
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C
2tan11+xx2+C
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D
None of these
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Solution

The correct option is A 2tan11xx+2+C
Given : dxx2x+2
Let : x2x+2=xt+2
x2x+2=x2t2+2+22xtx=(22t+1)(1+t2)dx=(22t2+2t22)dt(1+t2)2
dxx2x+2=(22t2+2t22)dt((22t+1)t(1+t2)+2)(1+t2)2

=(22t2+2t22)dt(22t2t+2+2t2)(1+t2)

=(22t2+2t22)dt(22t2t+2)(1+t2)

=2dt(1+t2)=2tan1(t)=2tan11xx+2+C

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