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Y'=4∙cos4x+2∙sin2x2+x+2dx
y'=4∙cos4x+2∙sin2x2+x+2dx=4∙cos4xdx+2∙sin2x2dx+x+2dx=1
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cos4xdx=cos2x=1+cos2∙x2=1+cos2∙x22dx=1+cos2∙x24dx=1+2∙cos2∙x+cos22∙x4dx=14∙dx+2∙cos2∙xdx+cos22∙xdx=14∙dx+cos2∙xd2∙x+1+cos2∙x2dx=14∙x+sin2∙x+12∙dx+cos4∙xdx=14∙x+sin2∙x+12∙x+44∙cos4∙xdx=14∙x+sin2∙x+x2+14∙cos4∙xd4∙x=14∙x+sin2∙x+x2+18∙sin4∙x+C∙x+sin2∙x+x2+18∙sin4∙x+C=14∙3∙x2+sin2∙x+18∙sin4∙x+C=3∙x8+sin2∙x4+sin4∙x32+C
sin2x2dx=sin2x=1-cos2∙x2=1-cosx2dx=12∙dx-cosxdx=12∙x-sinx+C=x2-sinx2+C
x+2dx=x+212dx+2=x+212+112+1+C=x+23232+C=23∙x+232+C
1=4∙3∙x8+sin2∙x4+sin4∙x32+2∙x2-sinx2+23∙x+232+C1=5∙x2+sin2∙x+sin4∙x8-sinx+23∙x+232+C1
24∙y'=60∙x+24∙sin2∙x+3∙sin4∙x-24∙sinx+16∙x+232+C1
24∙y=60∙x+24∙sin2∙x+3∙sin4∙x-24∙sinx+16∙x+232+C1dx=60∙xdx+24∙sin2∙xdx+3∙sin4∙xdx-24∙sinxdx+16∙x+232dx+C1∙dx==60∙xdx+12∙sin2∙xd2∙x+3∙44∙sin4∙xdx-24∙sinxdx+16∙x+232dx+C1∙dx=60∙x22-12∙cos2∙x-34∙cos4∙x+24∙cosx+16∙x+232+132+1+C1∙x+C2=
=30∙x2-12∙cos2∙x-34∙cos4∙x+24∙cosx+16∙x+25252+C1∙x+C2=30∙x2-12∙cos2∙x-34∙cos4∙x+24∙cosx+325∙x+252+C1∙x+C2
y=3024∙x2-1224∙cos2∙x-124∙34∙cos4∙x+cosx+124∙325∙x+252+C1∙x+C2=54∙x2-cos2∙x2-cos4∙x32+cosx+415∙x+252+C1∙x+C2
Ответ: y=54∙x2-cos2∙x2-cos4∙x32+cosx+415∙x+252+C1∙x+C2