Самый кондовый способ
![ \int\limits {\frac{dx}{\sqrt{e^{2x}+e^x+1}}} = \int\limits { \frac{dx}{\sqrt{(e^{x}+\frac{1}{2})^2+\frac{3}{4}}} } = \\\\ e^x+\frac{1}{2}=u\\ e^xdx=du\\\\ \int\limits{\frac{du}{(u-\frac{1}{2})\sqrt{u^2+\frac{3}{4}}}}\\\\ u=\frac{\sqrt{3}}{2}tga\\\\ du=\frac{\sqrt{3}da}{2cos^2a}\\\\ \int\limits {\frac{\frac{\sqrt{3}da}{2cos^2a}}{(\frac{\sqrt{3}tga}{2}-\frac{1}{2})\sqrt{\frac{3}{4}(tg^2a+1)}}}\\ ](https://tex.z-dn.net/?f=%0A%5Cint%5Climits+%7B%5Cfrac%7Bdx%7D%7B%5Csqrt%7Be%5E%7B2x%7D%2Be%5Ex%2B1%7D%7D%7D+%3D+%5Cint%5Climits+%7B+%5Cfrac%7Bdx%7D%7B%5Csqrt%7B%28e%5E%7Bx%7D%2B%5Cfrac%7B1%7D%7B2%7D%29%5E2%2B%5Cfrac%7B3%7D%7B4%7D%7D%7D+%7D+%3D+%5C%5C%5C%5C+e%5Ex%2B%5Cfrac%7B1%7D%7B2%7D%3Du%5C%5C+e%5Exdx%3Ddu%5C%5C%5C%5C+%5Cint%5Climits%7B%5Cfrac%7Bdu%7D%7B%28u-%5Cfrac%7B1%7D%7B2%7D%29%5Csqrt%7Bu%5E2%2B%5Cfrac%7B3%7D%7B4%7D%7D%7D%7D%5C%5C%5C%5C+u%3D%5Cfrac%7B%5Csqrt%7B3%7D%7D%7B2%7Dtga%5C%5C%5C%5C+du%3D%5Cfrac%7B%5Csqrt%7B3%7Dda%7D%7B2cos%5E2a%7D%5C%5C%5C%5C+%5Cint%5Climits+%7B%5Cfrac%7B%5Cfrac%7B%5Csqrt%7B3%7Dda%7D%7B2cos%5E2a%7D%7D%7B%28%5Cfrac%7B%5Csqrt%7B3%7Dtga%7D%7B2%7D-%5Cfrac%7B1%7D%7B2%7D%29%5Csqrt%7B%5Cfrac%7B3%7D%7B4%7D%28tg%5E2a%2B1%29%7D%7D%7D%5C%5C%0A)
Подставляя получаем
воспользуемся универсальной тригонометрической заменой
![sina=\frac{2t}{1+t^2}\\ cosa=\frac{1-t^2}{1+t^2}\\ da=\frac{2dt}{1+t^2}\\\\ \int\limits {\frac{\frac{4dt}{1+t^2}}{\sqrt{3}*\frac{2t}{1+t^2}-\frac{1-t^2}{1+t^2}}} =\\\\ \int\limits {\frac{4dt}{t^2+2\sqrt{3}t-1} }=\\\\ \int\limits{\frac{4dt}{(t+\sqrt{3})^2-4}} = |t+\sqrt{3}=z ; \ \ \ dt=dz\\\\ \int\limits{\frac{4dz}{z^2-4}}=ln(2-z)-ln(2+z)+C=\\\\ ln(2-t-\sqrt{3})-ln(2+t+\sqrt{3})+C=\\\\ ](https://tex.z-dn.net/?f=sina%3D%5Cfrac%7B2t%7D%7B1%2Bt%5E2%7D%5C%5C%0Acosa%3D%5Cfrac%7B1-t%5E2%7D%7B1%2Bt%5E2%7D%5C%5C%0Ada%3D%5Cfrac%7B2dt%7D%7B1%2Bt%5E2%7D%5C%5C%5C%5C%0A+%5Cint%5Climits+%7B%5Cfrac%7B%5Cfrac%7B4dt%7D%7B1%2Bt%5E2%7D%7D%7B%5Csqrt%7B3%7D%2A%5Cfrac%7B2t%7D%7B1%2Bt%5E2%7D-%5Cfrac%7B1-t%5E2%7D%7B1%2Bt%5E2%7D%7D%7D+%3D%5C%5C%5C%5C%0A+%5Cint%5Climits+%7B%5Cfrac%7B4dt%7D%7Bt%5E2%2B2%5Csqrt%7B3%7Dt-1%7D+%7D%3D%5C%5C%5C%5C%0A+%5Cint%5Climits%7B%5Cfrac%7B4dt%7D%7B%28t%2B%5Csqrt%7B3%7D%29%5E2-4%7D%7D+%3D+%7Ct%2B%5Csqrt%7B3%7D%3Dz+%3B+%5C+%5C+%5C++dt%3Ddz%5C%5C%5C%5C%0A+%5Cint%5Climits%7B%5Cfrac%7B4dz%7D%7Bz%5E2-4%7D%7D%3Dln%282-z%29-ln%282%2Bz%29%2BC%3D%5C%5C%5C%5C%0Aln%282-t-%5Csqrt%7B3%7D%29-ln%282%2Bt%2B%5Csqrt%7B3%7D%29%2BC%3D%5C%5C%5C%5C%0A)
Заменяя на
![t](https://tex.z-dn.net/?f=t)
и
![u](https://tex.z-dn.net/?f=u)
получаем
Ответ
Решение задания приложено
а) x^2 -121= (x-11) (x+11)
б)-0,04a^2+b^2c^2= - (0, 04-b^2c^2)= - (0,2-bc) (0,2+bc)
в)(x-1)^2-9= (x-1-3) (x-1+3)= (x-4) (x+2)
P=n*m + k(m+4) + L(2+m+4)=n*m + k(m+4) + L(6+m)