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x-x-2-5-1-2-dx-3-x-x-2-5-1-2-dx-x-x-2-5-3-x-2-5-dx-




Question Number 186310 by normans last updated on 03/Feb/23
     ((∫x(x^2 +5)^(1/2) dx − 3∫x(x^2 +5)^(−1/2)  dx)/(∫  ((x[(x^2 +5)−3])/( (√(x^2 +5  )))) dx)) =??
$$ \\ $$$$\:\:\:\frac{\int\boldsymbol{{x}}\left(\boldsymbol{{x}}^{\mathrm{2}} +\mathrm{5}\right)^{\mathrm{1}/\mathrm{2}} \boldsymbol{{dx}}\:−\:\mathrm{3}\int\boldsymbol{{x}}\left(\boldsymbol{{x}}^{\mathrm{2}} +\mathrm{5}\right)^{−\mathrm{1}/\mathrm{2}} \:\boldsymbol{{dx}}}{\int\:\:\frac{\boldsymbol{{x}}\left[\left(\boldsymbol{{x}}^{\mathrm{2}} +\mathrm{5}\right)−\mathrm{3}\right]}{\:\sqrt{\boldsymbol{{x}}^{\mathrm{2}} +\mathrm{5}\:\:}}\:\boldsymbol{{dx}}}\:=??\:\:\:\: \\ $$$$ \\ $$
Answered by Frix last updated on 03/Feb/23
=1+((3(C_1 −C_2 −C_3 ))/((x^2 −4)(√(x^2 +5))+3C_3 ))
$$=\mathrm{1}+\frac{\mathrm{3}\left(\mathrm{C}_{\mathrm{1}} −{C}_{\mathrm{2}} −{C}_{\mathrm{3}} \right)}{\left({x}^{\mathrm{2}} −\mathrm{4}\right)\sqrt{{x}^{\mathrm{2}} +\mathrm{5}}+\mathrm{3}{C}_{\mathrm{3}} } \\ $$

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