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[tex] \int\limits {(1-x+2x^2)sin4x} \, dx [/tex]
[tex] \int\limits {(1-x+2x^2)sin4x} \, dx [/tex]
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Int (1 - x + 2x^2)*sin 4x dx = Int sin 4x dx - Int x*sin 4x dx + 2*Int x^2*sin 4x dx == -1/4*cos 4x - |u=x, dv=sin 4x dx, du=dx, v=-1/4*cos 4x| +
+ |u=x^2, dv=sin 4x dx, du=2x dx, v=-1/4*cos 4x| =
= -1/4*cos 4x + x/4*cos 4x - 1/4*Int cos 4x dx -- x^2/2*cos 4x + Int x*cos 4x dx =
= -1/4*cos 4x + x/4*cos 4x - 1/16*sin 4x - x^2/2*cos 4x +
+ |u=x, dv=cos 4x dx, du=dx, v=1/4*sin 4x| =
= -1/4*cos 4x + x/4*cos 4x - 1/16*sin 4x - x^2/2*cos 4x +
+ x/4*sin 4x - 1/4*Int sin 4x dx =
= -1/4*cos 4x + x/4*cos 4x - 1/16*sin 4x - x^2/2*cos 4x +
+ x/4*sin 4x + 1/16*cos 4x + C