Series[Exp[Sin[3x^2]], {x, 0, 6}]

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 Š 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 Š 1, y@0D Š 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