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Question Number 38105 by maxmathsup by imad last updated on 21/Jun/18
find ∫     (dx/( (√(2x+1)) +(√(2x−1))))
$${find}\:\int\:\:\:\:\:\frac{{dx}}{\:\sqrt{\mathrm{2}{x}+\mathrm{1}}\:+\sqrt{\mathrm{2}{x}−\mathrm{1}}}\: \\ $$
Answered by MJS last updated on 22/Jun/18
∫(dx/( (√(2x+1))+(√(2x−1))))=(1/2)∫((√(2x+1))−(√(2x−1)))dx=  =(1/6)((2x+1)^(3/2) −(2x−1)^(3/2) )+C
$$\int\frac{{dx}}{\:\sqrt{\mathrm{2}{x}+\mathrm{1}}+\sqrt{\mathrm{2}{x}−\mathrm{1}}}=\frac{\mathrm{1}}{\mathrm{2}}\int\left(\sqrt{\mathrm{2}{x}+\mathrm{1}}−\sqrt{\mathrm{2}{x}−\mathrm{1}}\right){dx}= \\ $$$$=\frac{\mathrm{1}}{\mathrm{6}}\left(\left(\mathrm{2}{x}+\mathrm{1}\right)^{\frac{\mathrm{3}}{\mathrm{2}}} −\left(\mathrm{2}{x}−\mathrm{1}\right)^{\frac{\mathrm{3}}{\mathrm{2}}} \right)+{C} \\ $$

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