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2) ![\int\limits { \frac{x^2}{(4x^3+3)^3}} \, dx;t=4x^3;dt=12x^32dx; \int\limits { \frac{x^2}{(4x^3+3)^3}} \, dx;t=4x^3;dt=12x^32dx;](https://tex.z-dn.net/?f=+%5Cint%5Climits+%7B+%5Cfrac%7Bx%5E2%7D%7B%284x%5E3%2B3%29%5E3%7D%7D+%5C%2C+dx%3Bt%3D4x%5E3%3Bdt%3D12x%5E32dx%3B+)
![\int\limits { \frac{x^2}{(4x^3+3)^3}}\, dx= \frac{1}{12} \int\limits {(t+3)^{-3}} \, dt=- \frac{1}{24(t+3)^{2}}+C= \int\limits { \frac{x^2}{(4x^3+3)^3}}\, dx= \frac{1}{12} \int\limits {(t+3)^{-3}} \, dt=- \frac{1}{24(t+3)^{2}}+C=](https://tex.z-dn.net/?f=+%5Cint%5Climits+%7B+%5Cfrac%7Bx%5E2%7D%7B%284x%5E3%2B3%29%5E3%7D%7D%5C%2C+dx%3D+%5Cfrac%7B1%7D%7B12%7D+%5Cint%5Climits+%7B%28t%2B3%29%5E%7B-3%7D%7D+%5C%2C+dt%3D-+%5Cfrac%7B1%7D%7B24%28t%2B3%29%5E%7B2%7D%7D%2BC%3D+++)
![=- \frac{1}{24(4x^3+3)^2}+C= =- \frac{1}{24(4x^3+3)^2}+C=](https://tex.z-dn.net/?f=%3D-+%5Cfrac%7B1%7D%7B24%284x%5E3%2B3%29%5E2%7D%2BC%3D+++)
![\int\limits {x^4lnx} \, dx; \int\limits {x^4lnx} \, dx;](https://tex.z-dn.net/?f=+%5Cint%5Climits+%7Bx%5E4lnx%7D+%5C%2C+dx%3B+)
![du= \frac{dx}{x;} du= \frac{dx}{x;}](https://tex.z-dn.net/?f=du%3D+%5Cfrac%7Bdx%7D%7Bx%3B%7D+)
![v= \frac{x^5}{5}; v= \frac{x^5}{5};](https://tex.z-dn.net/?f=v%3D+%5Cfrac%7Bx%5E5%7D%7B5%7D%3B+)
![\int\limits {x^4lnx} \, dx= \frac{x^5}{5}lnx- \frac{1}{5}\int\limits {x^4} \, dx = \frac{1}{5}x^5lnx- \frac{1}{25}x^5+C; \int\limits {x^4lnx} \, dx= \frac{x^5}{5}lnx- \frac{1}{5}\int\limits {x^4} \, dx = \frac{1}{5}x^5lnx- \frac{1}{25}x^5+C;](https://tex.z-dn.net/?f=+%5Cint%5Climits+%7Bx%5E4lnx%7D+%5C%2C+dx%3D+%5Cfrac%7Bx%5E5%7D%7B5%7Dlnx-+%5Cfrac%7B1%7D%7B5%7D%5Cint%5Climits+%7Bx%5E4%7D+%5C%2C+dx+%3D+%5Cfrac%7B1%7D%7B5%7Dx%5E5lnx-+%5Cfrac%7B1%7D%7B25%7Dx%5E5%2BC%3B+)
![\int\limits { \frac{3x^2+8}{x^3+4x^2+4}} \, dx= \int\limits { \frac{3x^2+8}{x(x^2+4x+4)}}\, dx= \int\limits { \frac{3x^2+8}{x(x+2)^2}}\, dx \int\limits { \frac{3x^2+8}{x^3+4x^2+4}} \, dx= \int\limits { \frac{3x^2+8}{x(x^2+4x+4)}}\, dx= \int\limits { \frac{3x^2+8}{x(x+2)^2}}\, dx](https://tex.z-dn.net/?f=+%5Cint%5Climits+%7B+%5Cfrac%7B3x%5E2%2B8%7D%7Bx%5E3%2B4x%5E2%2B4%7D%7D+%5C%2C+dx%3D+%5Cint%5Climits+%7B+%5Cfrac%7B3x%5E2%2B8%7D%7Bx%28x%5E2%2B4x%2B4%29%7D%7D%5C%2C+dx%3D+%5Cint%5Climits+%7B+%5Cfrac%7B3x%5E2%2B8%7D%7Bx%28x%2B2%29%5E2%7D%7D%5C%2C+dx++)
![\frac{3x^2+8}{x(x+2)^2}= \frac{A}{x} + \frac{B}{x+2} + \frac{C}{(x+2)^2}= \frac{A(x+2)^2+Bx(x+2)+Cx}{x(x+2)^2}= \frac{3x^2+8}{x(x+2)^2}= \frac{A}{x} + \frac{B}{x+2} + \frac{C}{(x+2)^2}= \frac{A(x+2)^2+Bx(x+2)+Cx}{x(x+2)^2}=](https://tex.z-dn.net/?f=+%5Cfrac%7B3x%5E2%2B8%7D%7Bx%28x%2B2%29%5E2%7D%3D+%5Cfrac%7BA%7D%7Bx%7D+%2B+%5Cfrac%7BB%7D%7Bx%2B2%7D+%2B+%5Cfrac%7BC%7D%7B%28x%2B2%29%5E2%7D%3D+%5Cfrac%7BA%28x%2B2%29%5E2%2BBx%28x%2B2%29%2BCx%7D%7Bx%28x%2B2%29%5E2%7D%3D++)
![= \frac{A x^{2} +4Ax+4A+Bx^2+2Bx+Cx}{x(x+2)^2}= \frac{(A+B) x^{2} +(4Ax++2B+C)x+4A}{x(x+2)^2}; = \frac{A x^{2} +4Ax+4A+Bx^2+2Bx+Cx}{x(x+2)^2}= \frac{(A+B) x^{2} +(4Ax++2B+C)x+4A}{x(x+2)^2};](https://tex.z-dn.net/?f=%3D+%5Cfrac%7BA+x%5E%7B2%7D+%2B4Ax%2B4A%2BBx%5E2%2B2Bx%2BCx%7D%7Bx%28x%2B2%29%5E2%7D%3D+%5Cfrac%7B%28A%2BB%29+x%5E%7B2%7D+%2B%284Ax%2B%2B2B%2BC%29x%2B4A%7D%7Bx%28x%2B2%29%5E2%7D%3B+)
![\left \{ {{B=1}\atop {4Ax+2B+C=0}}\atop {4A=8}} \right.; \left \{ {{A+B=3}\atop {C=-10}}\atop {A=2}} \right \left \{ {{B=1}\atop {4Ax+2B+C=0}}\atop {4A=8}} \right.; \left \{ {{A+B=3}\atop {C=-10}}\atop {A=2}} \right](https://tex.z-dn.net/?f=+%5Cleft+%5C%7B+%7B%7BB%3D1%7D%5Catop+%7B4Ax%2B2B%2BC%3D0%7D%7D%5Catop+%7B4A%3D8%7D%7D+%5Cright.%3B++%5Cleft+%5C%7B+%7B%7BA%2BB%3D3%7D%5Catop+%7BC%3D-10%7D%7D%5Catop+%7BA%3D2%7D%7D+%5Cright+)
![\frac{3x^2+8}{x(x+2)^2}= \frac{2}{x} + \frac{1}{x+2} - \frac{10}{(x+2)^2} \frac{3x^2+8}{x(x+2)^2}= \frac{2}{x} + \frac{1}{x+2} - \frac{10}{(x+2)^2}](https://tex.z-dn.net/?f=+%5Cfrac%7B3x%5E2%2B8%7D%7Bx%28x%2B2%29%5E2%7D%3D+%5Cfrac%7B2%7D%7Bx%7D+%2B+%5Cfrac%7B1%7D%7Bx%2B2%7D+-+%5Cfrac%7B10%7D%7B%28x%2B2%29%5E2%7D)
![\int\limits { \frac{3x^2+8}{x^3+4x^2+4}} \, dx= \int\limits { \frac{3x^2+8}{x(x+2)^2}}\, dx= \int\limits { \frac{2}{x}}\, dx+ \int\limits { \frac{1}{x+2}}\, dx- \int\limits { \frac{10}{(x+2)^2}}\, dx= \int\limits { \frac{3x^2+8}{x^3+4x^2+4}} \, dx= \int\limits { \frac{3x^2+8}{x(x+2)^2}}\, dx= \int\limits { \frac{2}{x}}\, dx+ \int\limits { \frac{1}{x+2}}\, dx- \int\limits { \frac{10}{(x+2)^2}}\, dx=](https://tex.z-dn.net/?f=%5Cint%5Climits+%7B+%5Cfrac%7B3x%5E2%2B8%7D%7Bx%5E3%2B4x%5E2%2B4%7D%7D+%5C%2C+dx%3D+%5Cint%5Climits+%7B+%5Cfrac%7B3x%5E2%2B8%7D%7Bx%28x%2B2%29%5E2%7D%7D%5C%2C+dx%3D+%5Cint%5Climits+%7B+%5Cfrac%7B2%7D%7Bx%7D%7D%5C%2C+dx%2B+%5Cint%5Climits+%7B+%5Cfrac%7B1%7D%7Bx%2B2%7D%7D%5C%2C+dx-+%5Cint%5Climits+%7B+%5Cfrac%7B10%7D%7B%28x%2B2%29%5E2%7D%7D%5C%2C+dx%3D)
![2 \int\limits { \frac{1}{x}}\, dx+ \int\limits { \frac{1}{x+2}}\, dx-10 \int\limits { \frac{1}{(x+2)^2}}\, dx=2lnx+ln(x+2)+ \frac{10}{x+2}+C; 2 \int\limits { \frac{1}{x}}\, dx+ \int\limits { \frac{1}{x+2}}\, dx-10 \int\limits { \frac{1}{(x+2)^2}}\, dx=2lnx+ln(x+2)+ \frac{10}{x+2}+C;](https://tex.z-dn.net/?f=2+%5Cint%5Climits+%7B+%5Cfrac%7B1%7D%7Bx%7D%7D%5C%2C+dx%2B+%5Cint%5Climits+%7B+%5Cfrac%7B1%7D%7Bx%2B2%7D%7D%5C%2C+dx-10+%5Cint%5Climits+%7B+%5Cfrac%7B1%7D%7B%28x%2B2%29%5E2%7D%7D%5C%2C+dx%3D2lnx%2Bln%28x%2B2%29%2B+%5Cfrac%7B10%7D%7Bx%2B2%7D%2BC%3B)
3)
![y'=\frac{2}{5}(x+2)^{- \frac{3}{5}}* \sqrt{x-4}*(x^2-1)^3+ y'=\frac{2}{5}(x+2)^{- \frac{3}{5}}* \sqrt{x-4}*(x^2-1)^3+](https://tex.z-dn.net/?f=y%27%3D%5Cfrac%7B2%7D%7B5%7D%28x%2B2%29%5E%7B-+%5Cfrac%7B3%7D%7B5%7D%7D%2A+%5Csqrt%7Bx-4%7D%2A%28x%5E2-1%29%5E3%2B)
![+\sqrt[5]{(x+2)^2}*\frac{1}{2\sqrt{x-4}}*(x^2-1)^3+ \sqrt[5]{(x+2)^2}* \sqrt{x-4}* 3(x^2-1)^2*2x= +\sqrt[5]{(x+2)^2}*\frac{1}{2\sqrt{x-4}}*(x^2-1)^3+ \sqrt[5]{(x+2)^2}* \sqrt{x-4}* 3(x^2-1)^2*2x=](https://tex.z-dn.net/?f=%2B%5Csqrt%5B5%5D%7B%28x%2B2%29%5E2%7D%2A%5Cfrac%7B1%7D%7B2%5Csqrt%7Bx-4%7D%7D%2A%28x%5E2-1%29%5E3%2B+%5Csqrt%5B5%5D%7B%28x%2B2%29%5E2%7D%2A+%5Csqrt%7Bx-4%7D%2A+3%28x%5E2-1%29%5E2%2A2x%3D)
![=\frac{(x^2-1)^2}{ \sqrt[]{(x+2)^3}\sqrt{x-4}}(\frac{2}{5}* (x-4)*(x^2-1)+ =\frac{(x^2-1)^2}{ \sqrt[]{(x+2)^3}\sqrt{x-4}}(\frac{2}{5}* (x-4)*(x^2-1)+](https://tex.z-dn.net/?f=%3D%5Cfrac%7B%28x%5E2-1%29%5E2%7D%7B+%5Csqrt%5B%5D%7B%28x%2B2%29%5E3%7D%5Csqrt%7Bx-4%7D%7D%28%5Cfrac%7B2%7D%7B5%7D%2A+%28x-4%29%2A%28x%5E2-1%29%2B)
![+\frac{1}{2} (x+2)*(x^2-1)+6x (x+2)*(x-4)); +\frac{1}{2} (x+2)*(x^2-1)+6x (x+2)*(x-4));](https://tex.z-dn.net/?f=%2B%5Cfrac%7B1%7D%7B2%7D+%28x%2B2%29%2A%28x%5E2-1%29%2B6x+%28x%2B2%29%2A%28x-4%29%29%3B)
![y=cos(cosx); y=cos(cosx);](https://tex.z-dn.net/?f=y%3Dcos%28cosx%29%3B)
![y'=-sin(cosx)*(-sinx)=sin(cosx)*sinx; y'=-sin(cosx)*(-sinx)=sin(cosx)*sinx;](https://tex.z-dn.net/?f=y%27%3D-sin%28cosx%29%2A%28-sinx%29%3Dsin%28cosx%29%2Asinx%3B)
3)
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