Alimentacion.

1. ft{ -π (-π<t<0)t(t<0<π)
a0 = -ππft
-π0-π dt+0π-π dt= -πt -π0+t220π
=π2+π22-π22 = π22 = π
an = 1π-ππcosntdt
=1π-π0-πcosntdt+0πtcosnt dt u=tdv=cos(nt)dt
du=dt v=1n sen (nt)
=1π -πnsen (nt)-π0+ tnsen nt- 0π1nsen ntdt
=1π -πnsen 0t+-πnsen πt + tnsen nt+1ncos ntdt 0π
=1π 1n2cos(nt)= 1πn2

bn = 1π-ππsenntdt
=1π-tncosntdt- 0π-1ncosnt dt0π
=1 π-tncosntdt+ 1n2sen nt0π
=1π 1n2cos(nt)= -2πn

π2+1πcost+ cos2t4+cos3t9+…+ cosntn2-2 sen t+ sen 2t2+sen 3t3+…+ sen ntn

2. ft{ t+π (-π<t<0)0(t<0<π)
a0=-ππt+π dt+ -ππ0 dt
=-π0t dt+ -π0πdt= t22+πt-π0= 0- π22+π2= -3π22
an = 1π-π0t+πcosntdt+0π0cosntdt =1π-π0tcosntdt+-π0πcosnt dt u=t dv=cos(nt)dtdu=dt v=1n sen (nt)
=1π tnsen nt--π0sen ntdt+ πnsen (nt)-π0
=1π tnsen nt+1n2cos πt+πnsen nt -π0
=1π 1n2cos n0- 1n2cos nπ= 1π2n2= 2πn2
bn = 1π-π0t+πsenntdt+0πsen nt dt
=1π-π0t sen nt dt+-π0π sennt dt u=t dv=sen nt dt
du=dt v=-1n cos (nt)
=1π -tncos nt--π0-1ncis ntdt- πncos (nt)-π0
=1π -tncos nt+1n2sen πt-πncos nt -π0=1π -0ncos n0- 1n2sen nπ- πncos n0=--πncos nπ+ 1n2sen nπ- -πncos nπ
=1π-πn--πn-πn=1π-πn--2πn=1ππn=1n

-3π2+3πcost+ cos2t4+cos3t9+…+ cosntn2+sen t+ sen 2t2+sen 3t3+…+ sen ntn

3. ft{1- tπ(0<t<2π)
a0 =02π1-tπsen nt dt
=02π dt-02πtn dt = t-1nt2202π= t-t22n02π
=2π-2π22n=2π-4π22n=2-2πn
an = 1π02π1-tncosntdt=1π02πcosntdt-02πtncosnt dt u=tdv=1ncos(nt)dt du=dt v=-1n2 sen (nt)
=1π 1nsen nt-tn2sen nt-1n3cos nt02π
=1π 1nsen 2π+2πn2sen 2π-1n3cos 2π-1nsen 0+2πn2sen 0-1n3cos 0
=1π-1n3--1n3=1πn2
bn = 1π02π1-tnsenntdt
=1π02πsenntdt-02πtnsenntdt= 1π -1ncos nt+tncos nt-02π-1ncosntdt02π
=1π -1ncos nt+tn2sen πt+1n3sen nt 02π=1π -1ncos 2π+2πn2cos 2π+1n3sen 2π--1ncos 0+ 1ncos 0+1n3sen 0
=1π-1n+2πn2--1n+1n=-1πn+2ππn2...