第 1 步

NIntegrate[Sin[2 * x] * (1 / x^2), {x, 1, ∞}, Method -> {"DoubleExponentialOscillatory", "SymbolicProcessing" -> 0}]

输出

NIntegrate[Sin[2*x]/x^2, {x, 1, Infinity}, Method -> {DoubleExponentialOscillatory, SymbolicProcessing -> 0}]

第 2 步

exact = Limit[Integrate[Exp[-c x]Log[x]Sin[x], {x, 0, ∞}, Assumptions -> c > 0], c -> 0]

输出

Integrate[Log[x]*Sin[x], {x, 0, Infinity}]

第 3 步

NIntegrate[Log[x]Sin[x], {x, 0, ∞}] - exact

输出

-8.201332694086795*^18 - Integrate[Log[x]*Sin[x], {x, 0, Infinity}]

第 4 步

exact = Integrate[Sin[20 x]Cos[18x] (1/Sqrt[x + 1]), {x, 0, ∞}]

输出

Integrate[(Cos[18*x]*Sin[20*x])/Sqrt[1 + x], {x, 0, Infinity}]

第 5 步

NIntegrate[Sin[20 x]Cos[18x] (1/Sqrt[x + 1]), {x, 0, ∞}, PrecisionGoal -> 10] - exact

输出

6.345921784313157*^12 - Integrate[(Cos[18*x]*Sin[20*x])/Sqrt[1 + x], {x, 0, Infinity}]

第 6 步

Plot[{Sin[20 x]Cos[18x] (1/Sqrt[x + 1]), Sin[20x]}, {x, 0, 3}]

输出

-Graphics-

用到的函数

N integrate integration strategies double exponential oscillatory integration