Series@Exp@Sin@3 x ^ 2DD, 8x, 0, 6<D 2 1+3x + 9 x4 + O@xD7 2 Series@Exp@Sin@4 x ^ 2DD, 8x, 0, 6<D 1 + 4 x2 + 8 x4 + O@xD7 Series@Exp@Sin@5 x ^ 2DD, 8x, 0, 6<D 2 1+5x + 25 x4 + O@xD7 2 Series@Exp@Sin@6 x ^ 2DD, 8x, 0, 6<D 1 + 6 x2 + 18 x4 + O@xD7 Integrate@HExp@xD * Log@1 + Exp@xDDL H1 + Exp@xDL, 1 Log@1 + ãxD2 2 Integrate@HExp@xD * Log@1 + Exp@xDDL H1 + Exp@xDL, 1 2 J-Log@2D2 + LogA1 + ã2E N 2 Integrate@HExp@xD * Log@1 + Exp@xDDL H1 + Exp@xDL, 1 2 J-Log@1 + ãD2 + LogA1 + ã2E N 2 Integrate@HExp@xD * Log@1 + Exp@xDDL H1 + Exp@xDL, 1 2 J-Log@2D2 + LogA1 + ã3E N 2 Integrate@HExp@xD * Log@1 + Exp@xDDL H1 + Exp@xDL, 1 2 J-Log@1 + ãD2 + LogA1 + ã3E N 2 Limit@H3 x - 6L Sin@6 Pi xD, x ® 2D 1 2Π Limit@H3 x - 9L Sin@4 Pi xD, x ® 3D 3 4Π Limit@H3 x - 12L Sin@2 Pi xD, x ® 4D 3 2Π Limit@H5 x - 25L Sin@5 Pi xD, x ® 5D 1 Π xD 8x, 0, 2<D 8x, 1, 2<D 8x, 0, 3<D 8x, 1, 3<D 2 luglio.nb D@x * Exp@1 Hx - 1LD, xD 1 ã 1 -1+x ã -1+x x - H-1 + xL2 Simplify@%D 1 ã -1+x I1 - 3 x + x2M H-1 + xL2 Simplify@D@D@x * Exp@1 Hx - 1LD, xD, xDD 1 ã -1+x H-2 + 3 xL H-1 + xL4 SolveA1 - 3 x + x2 0, xE 1 I3 - 5 M>, :x ® 1 I3 + 5 M>> 2 2 Plot@x * Exp@1 Hx - 1LD, 8x, -5, 5<D ::x ® 15 10 5 -4 -2 2 4 Limit@Exp@1 Hx - 1LD, x ® InfinityD 1 Limit@x Exp@1 Hx - 1LD - x, x ® InfinityD 1 Integrate@H4 x ^ 2 + 5 x + 6L Hx ^ 2 + 2 x + 2L, xD 3 4 x + ArcTan@1 + xD - LogA2 + 2 x + x2E 2 DSolve@8y ''@xD - 4 y '@xD + 4 y@xD == Exp@2 xD, y '@0D 1, y@0D 1<, y@xD, xD 1 ::y@xD ® ã2 x I2 - 2 x + x2M>> 2 DSolve@y ''@xD - 4 y '@xD + 4 y@xD == Exp@2 xD, y@xD, xD 1 2x 2 ::y@xD ® ã x + ã2 x C@1D + ã2 x x C@2D>> 2