1) [latex] intlimits { frac{1}{x(lnx+3)} } , dx= intlimits { frac{1}{lnx+3} } , d(lnx) = ln|lnx+3|+C [/latex] dlatego bo : [latex]d(lnx)= frac{dx}{x} [/latex] 2) [latex] intlimits { frac{cosx+1}{1+sinx} } , dx = intlimits { frac{(cosx+1)(1-sinx)}{(1+sinx)(1-sinx)} } , dx = \ \ =intlimits { frac{cosx-sinxcosx+1-sinx}{cos^2x} } , dx= \ \ = intlimits { frac{dx}{cosx} }- intlimits { frac{sinx}{cosx} } , dx + intlimits { frac{dx}{cos^2x} }- intlimits { frac{sinx}{cos^2x} } , dx = \ \ = intlimits{ frac{dx}{cosx} }+ln|cosx|+tgx+ intlimits { cos^{-2}x } , d(cosx) = ...\ \ [/latex] pomocnicza całka [latex] intlimits { frac{1}{cosx} } , dx= intlimits { frac{cosx}{1-sin^2x} } , dx= intlimits { frac{1}{(1+sinx)(1-sinx)} } , d(sinx)=|_{t=sinx} = \ \ = intlimits { frac{1}{(1+t)(1-t)} } , dt= frac{1}{2} ln( frac{1+t}{1-t} )+C|_{t=sinx} = frac{1}{2} ln( frac{1+sinx}{1-sinx} )+C[/latex] pomocnicza całka [latex]- intlimits { frac{sinx}{cos^2x} } , dx= intlimits {cos^{-2}x} , d(cosx) = - frac{1}{cosx}+C [/latex] ODP ostatecznie [latex] intlimits { frac{cosx+1}{sinx+1} } , dx= frac{1}{2} ln( frac{1+sinx}{1-sinx} )+ln|cosx|+tgx- frac{1}{cosx}+C [/latex] 3) [latex] intlimits { frac{3^x}{ sqrt{1-9^x} } } , dx= egin{vmatrix} 3^x=t\3^xln3dx=dt\3^xdx= frac{dt}{ln3} end{vmatrix}= \ \ = frac{1}{ln3} intlimits { frac{dt}{ sqrt{1-t^2} } }= frac{1}{ln3}arcsint+C= frac{arcsin(3^x)}{ln3}+C [/latex] 2*) alternatywa z podstawieniem [latex]t=tg frac{x}{2} [/latex] . [latex] intlimits{ frac{cosx+1}{sinx+1} } , dx= egin{vmatrix} t=tg frac{x}{2} \sinx= frac{2t}{1+t^2} \cosx= frac{1-t^2}{1+t^2} \ dx= frac{2dt}{1+t^2} end{vmatrix}=... 4intlimits { frac{1}{(1+t^2)(t+1)^2} } , dt=... \ \ \ frac{(t+1)ln(t+1)-2-(t+1)ln(t^2+1)}{t+1} |_{t=tg frac{x}{2} }=... [/latex] [latex] frac{2(t+1)ln(t+1)-(t+1)ln(t^2+1)-2}{t+1}=2ln(t+1)-ln(t^2+1)- frac{2}{t+1} =(*) \ \ t=tg( frac{x}{2} ) \ \ (*)=2ln(tg( frac{x}{2} )+1)-ln(tg^2( frac{x}{2})+1 ))- frac{2}{tg( frac{x}{2} )+1} [/latex]
