Question

Помогите решить, 45 баллов.

Answer 1

#1:
$intlimits{ frac{1}{ sqrt{2x-1}- sqrt[4]{2x-1} } } , dx=\| 2x-1=t^4\| 2 dx=4t^3 dt\| dx=2t^3 dt\| t= sqrt[4]{2x-1}\ intlimits { frac{2t^3}{t^2-t} } , dt= 2intlimits { frac{t^2}{t-1} } , dt=2 (intlimits { frac{(t-1)(t+1)}{t-1} } , dt+ intlimits{ frac{1}{t-1} } , dt)=2( intlimits {t} , dt+\ +intlimits {1} , dt+ intlimits {( t-1)^{-1} } , dt)\=2( frac{t^2}{2}+t+ln|t-1|)=t^2+2t+2ln|t-1|\ sqrt{2x-1}+ 2sqrt[4]{2x-1}+2ln|sqrt[4]{2x-1}-1|+C$

#2:
$intlimits { frac{1}{x sqrt{4+x^2}} } , dx=\| 4+x^2=t^2 ; x^2=t^2-4 \| 2x dx=2t dt\| dx= frac{t}{x} dt\| t= sqrt{4+x^2} \ intlimits { frac{t}{x^2(t)} } , dt= intlimits { frac{1}{t^2-2^2} } , dt= frac{1}{4}ln| frac{t-2}{t+2}|+C\ frac{1}{4}ln| frac{ sqrt{4+x^2}-2 }{ sqrt{4+x^2}+2 }|+C$