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Question

# The solution of ∫dx√x2+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 log∣x+√x2+c∣We have to find the solution of ∫dx√x2+c using Euler substitution.Consider ∫dx√x2+cWe can use Euler first substitution: √x2+c=−x+t ⇒x2+c=(−x+t)2 ⇒x2+c=x2−2xt+t2 ⇒x=t2−c2tDifferentiating both sides we getdx=t2+c2t2dtAlso √x2+c=−t2−c2t+t=t2+c2t∴∫dx√x2+c=∫t2+c2t2dtt2+c2t =∫dtt =log|t| =log|x+√x2+c|Hence ∫dx√x2+c=log|x+√x2+c|

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