mathematica练习程序（第三章 微积分）

习题：

代码如下：

1.
Limit[(Exp[x] - Exp[-x])/Sin[x], x -> 0]

Limit[(x/(x - 1)) - 1/Log[x], x -> 1]

Limit[(1 - Cos[x])/x^2, x -> 0]

Limit[(1 + 1/x^2)^x, x -> Infinity]

2.
D[a^x Log[x], x]

D[(1 - Log[x])/(1 + Log[x]), x]

D[(1 + (1 + x^(1/3))^(1/3))^(1/3), x]

D[ArcTan[(1 + x)/(1 - x)], x]

D[Exp[x] + Exp[Exp[x]], x]

D[x^(x^x), x]

D[(Sin[x])^Cos[x], x]

D[Log[Cos[ArcTan[(Exp[x] - Exp[-x])/2]]], x]

3.
D[Sin[x] Sin[2 x] Sin[3 x], {x, 20}]

D[ArcTan[x], {x, 20}]

D[1/(1 - x^2), {x, 60}]

D[(1 + x)/Sqrt[1 - x], {x, 60}]

4.
Integrate[(2 x - 3)^100, x]

Integrate[(x + 1)/Sqrt[x], x]

Integrate[x^2 a^x, x]

Integrate[(2 x^2 - 5)/(x^4 - 5 x^2 + 6), x]

Integrate[Log[x + Sqrt[1 + x^2]], x]

Integrate[(Exp[2 x] + 1)/(Exp[x] + 1), x]

Integrate[ArcTan[y, x], x, y]

Integrate[x y z (1 - x - y), x, y, z]

5.
Integrate[Sin[x]^2 Cos[x]^2, {x, 0, 1}]

Integrate[Sqrt[Exp[x] - 1], {x, 0, Log[2]}]

Integrate[Sqrt[Exp[x]]/Sqrt[Exp[x] + Exp[-x]], {x, 0, 1}]

Integrate[x^2/Sqrt[x^2 + a^2], {x, 0, a}]

Integrate[x^2 + y^3, {x, 1, 2}, {y, 1, 1 - x}]

Integrate[x y^2, {x, 0, 1}, {y, x^2, x}]

Integrate[r^2 Sin[y]^2, {y, 0, 2 Pi}, {r, 0, a}]

Integrate[x y z, {x, 0, 1}, {y, 0, x}, {z, 0, x + y}]

6.
Series[Exp[x^2], {x, 0, 5}]

Series[x^2/(1 - x), {x, 0, 5}]

Series[Log[Sqrt[(1 + x)/(1 - x)]], {x, 0, 5}]

Series[(1 + x) Exp[-x], {x, 0, 5}]

Series[Cos[x] Cos[y], {x, 0, 5}, {y, 0, 5}]

7.
DSolve[{Derivative[1][y][x] == x + y[x], y[0] == 1}, y[x], x]

DSolve[{x'[t] + y[t] == Cos[t], y'[t] + x[t] == Sin[t]}, {y[t], x[t]},
  t]

DSolve[{x'[t] == 2 x[t] - y[t] + z[t], 
  y'[t] == 2 x[t] + 2 y[t] - z[t], 
  z'[t] == x[t] + 2 y[t] - z[t]}, {x[t], y[t], z[t]}, t]