Séries de Fourier - Soluções Detalhadas de Exercícios Resolvidos

Encontre a série de Fourier da função dada Soluções Questao 013o fx x L x L fx 2L fx 1 serie fx a02 sum n1 to infinity an cos nπxL bn sen nπxL 2 a0 a0 12 L to L fx dx a0 12 L to L x dx 1L x22 LL a0 1L L22 L22 0 a0 0 3 an an 12 L to L fx cos nπxL dx an 12 L to L x cos nπxL dx i integrando por partes metodo tabular D I x cos nπxL 1 sen nπxL Lnπ 0 cos nπxL L2n2 π2 an 12 x sen nπxL Lnπ cos nπxL L2n2 π2 LL an 1nπ L cos nπ L cos nπ bn 2Lnπ cos nπ 2Lnπ 1n 4 Dn bn 12 L to L fx sen nπxL dx bn 12 L to L x sen nπxL dx ii metodo tabular integracao por partes D I x sen nπxL 1 cos nπxL Lnπ 0 sen nπxL L2n2 π2 bn 12 x cos nπxL Lnπ sen nπxL L2n2 π2 LL bn 1nπ 1 cos nπ bn 1n 1nπ 5 fx a02 sum n1 to infinity an cos nπxL bn sen nπxL fx sum n1 to infinity 2Lnπ 1n sen nπxL fx 2Lπ sum n1 to infinity 1nn sen nπxL Fourier Questao 014o fx 1 2 x 0 0 0 x 2L fx 2L fx 1 a0 a0 L to L fx dx a0 L to 0 1 dx 0 to L 0 dx a0 x L to 0 L 2 a0 1π x22 0 to π π2 2 an an 1L L to 0 1 cos nπxL dx 1L 0 to L 0 dx an 12 sen nπxL Lnπ LL an 0 3 bn bn 1L L to 0 sen nπxL dx 1L 0 to L 0 dx bn 1L cos nπxL Lnπ 0 to L bn 1nπ 1 cos nπ 1n 1 nπ bn 1n 1nπ bn0 n par bn 2nπ n ímpar 4 serie fx a02 sum n1 to infinity an cos nπxL bn sen nπxL fx L2 sum n1 to infinity 22n1π sen 2n1πxL fx L2 2π sum n1 to infinity sen 2n1 π xL 2n1 Questao 015o fx x π x 0 0 0 x π fx 2π fx 1 a0 a0 12 π to 0 x dx 12 0 to π 0 dx 2L 2π L π a0 1π x22 0 to π π2 2 an an 1π π to 0 x cos nx dx 0 to π 0 dx 1 Integracao por partes metodo tabular D I x cosnx sennx 1n 0 cosnx 1n2 an 1π x sennx n cos nx n2 0 to π an 1π 1n2 cos nπ n2 an Lπ n2 1 1n an 0 n par an 2π n ímpar 3 bn 1π π to 0 x sen nx dx 0 to π 0 dx D I x sennx cosnx 1n 0 sennx 1n2 bn 1π x cosnx n sennx n2 0 to π bn 1π πn cos nπ bn 1n1 n 4 fx a02 sum n1 to infinity an cos nπxL bn sen nπx L fx π4 sum n1 to infinity 2 cos 2n1x π 2n12 1n1 n sen nx Questao 016 fx x1 1 x 0 1x 0 x 1 fx2 fx 1 a0 2L 2 L 1 a0 12 from 1 to 0 of x1 dx 12 from 0 to 1 of 1x dx a0 x²2 x from 1 to 0 x x²2 from 0 to 1 a0 12 1 1 12 2 1 1 2 an an from 1 to 0 x1cosnπx dx from 0 to 1 1xcosnπx dx i Integral por partes D I x1 cosnπx 1 sennπxnπ 0 cosnπxn²π² 1 1 x cosnπxnπ 1 sennπxn²π² 0 0 an x1sennπxnπ cosnπxn²π² from 1 to 0 1xsennπxnπ cosnπxn²π² from 0 to 1 an 1n²π² 1nn²π² 1nn²π² 1n²π² an 2n²π²1 1n an0 n par an 4n²π² n impar 3 bn bn from 1 to 0 of x1 sennπx dx from 0 to 1 1x sennπx dx D I x1 sennπx 1 cosnπxnπ 1 sennπx 1n²π² 0 D I 1x sennπx 1 cosnπxnπ 1 sennπx 1n²π² 0 bn x1cosnπxnπ sennπxn²π² from 1 to 0 x1cosnπxnπ sennπxn²π² from 0 to 1 bn 1nπ 1nπ 0 4 fx a02 from n1 to an cosnπxL bn sennπxL fx 12 from n1 to 42n1²π²cos2n1πx fx 12 4π² from n1 to cos2n1πx 2n1² Fourier Questao 017 fx xL L x 0 L 0 x L fx2L fx 1 a0 a0 12 from L to 0 of xL dx 1L from 0 to L of L dx a0 12 x²2 Lx from L to 0 12 Lx from 0 to L a0 12 L²2 L² 12 L² a0 L2 L 3L2 2 an an 1L from L to 0 of xLcosnπxL dx 1L from 0 to L of L cosnπxL dx D I xL cosnπxL L sennπxLnπ 0 cosnπxL L²n²π² 0 D I L cosnπxL sennπxLnπ 0 an 1L xLsennπxLLnπ cosnπxL L²n²π² from L to 0 L sennπxLLnπ from 0 to L an 1n²π² Ln²π² cosnπ Ln²π² 1 1n an0 n par an2Ln²π² n impar an0 n par an 2Ln²π² n impar 3 bn bn 12 from L to 0 xLsennπxL dx 12 from 0 to L sennπxL dx D I xL sennπxL 1 cosnπxL Lnπ 0 sennπxL L²n²π² 0 bn 12 xLcosnπxLLnπ sennπxLL²n²π² from L to 0 cosnπxLLnπ from 0 to L bn Lnπ cosnπLnπ Lnπ bn 1n1nπ 4 fx fx a02 from n1 to an cosnπxL bn sennπxL fx 3L4 from n1 to 2L2n1²π²cos2n1πxL 1n1nπ sinnπxL Questao 018 fx 0 2 x 1 x 1 x 1 0 1 x 2 fx4 fx 1 a0 2L 4 L 2 a0 12 from 2 to 0 of 0 dx 12 from 1 to 1 x dx 1L from 1 to 2 of 0 dx a0 12 x²2 from 1 to 1 0 2 an an 12 from 2 to 1 of 0 dx 12 from 1 to 1 x cosnπx2 dx 12 from 1 to 2 of 0 dx i Integral por partes D I x cosnπx2 1 sennπx2 2nπ 0 cosnπx2 4n²π² an x sennπx2 2nπ cosnπx2 4n²π² from 1 to 1 an 0 3 bn bn 12 from 2 to 0 of 0 dx 12 from 1 to 1 of x sennπx2 dx from 1 to 2 of 0 dx bn 12 from 1 to 1 x sennπx2 dx D I x sennπx2 0 cosnπx2 2nπ 1 sennπx2 4n²π² bn 12 x cosnπx2 2nπ sennπx2 4n²π² from 1 to 1 bn cosnπ2 1nπ cosnπ2 1nπ sennπ2 2n²π² sennπ2 2n²π² bn 2nπ cosnπ2 2nπ² sennπ2 4 fx fx a02 from n1 to an cosnπx2 bn sennπx2 fx from n1 to 2nπ cosnπx2 2nπ² sennπx2 sennπx2