Sympy 积分的分步解法

问题描述

在 sympy 的文档中 http://docs.sympy.org/latest/modules/integrals/integrals.html 我们可以阅读:

In doc of sympy http://docs.sympy.org/latest/modules/integrals/integrals.html we can read:

manualintegrate 模块具有返回所使用步骤的函数(有关更多信息，请参阅模块文档字符串).

但是调用 help(sympy.integrals.manualintegrate) 我们得到:

but calling help(sympy.integrals.manualintegrate) we get:

Help on function manualintegrate in module sympy.integrals.manualintegrate:

manualintegrate(f, var)
manualintegrate(f, var)

Compute indefinite integral of a single variable using an algorithm that
resembles what a student would do by hand.

Unlike ``integrate``, var can only be a single symbol.

Examples
========

>>> from sympy import sin, cos, tan, exp, log, integrate
>>> from sympy.integrals.manualintegrate import manualintegrate
>>> from sympy.abc import x
>>> manualintegrate(1 / x, x)
log(x)
>>> integrate(1/x)
log(x)
>>> manualintegrate(log(x), x)
x*log(x) - x
>>> integrate(log(x))
x*log(x) - x
>>> manualintegrate(exp(x) / (1 + exp(2 * x)), x)
atan(exp(x))
>>> integrate(exp(x) / (1 + exp(2 * x)))
RootSum(4*_z**2 + 1, Lambda(_i, _i*log(2*_i + exp(x))))
>>> manualintegrate(cos(x)**4 * sin(x), x)
-cos(x)**5/5
>>> integrate(cos(x)**4 * sin(x), x)
-cos(x)**5/5
>>> manualintegrate(cos(x)**4 * sin(x)**3, x)
cos(x)**7/7 - cos(x)**5/5
>>> integrate(cos(x)**4 * sin(x)**3, x)
cos(x)**7/7 - cos(x)**5/5
>>> manualintegrate(tan(x), x)
-log(cos(x))
>>> integrate(tan(x), x)
-log(sin(x)**2 - 1)/2

See Also
========

sympy.integrals.integrals.integrate
sympy.integrals.integrals.Integral.doit
sympy.integrals.integrals.Integral

我没有看到逐步的解决方案.

I don't see step by step solution.

推荐答案

您正在查看 function manualintegrate 的文档字符串，而不是 module manualintegrate 的文档字符串.该模块是这里，它说

You are looking at the docstring of the function manualintegrate, not of the module manualintegrate. The module is here and it says

该模块还提供了获取用于评估评估的步骤的功能特殊积分，在 integral_steps 函数中.这将返回表示使用的集成规则的嵌套命名元组.

integral_steps 函数记录如下:

返回计算积分所需的步骤.此功能试图尽可能地反映学生将用手做的事情.SymPy Gamma 使用它来提供积分的逐步解释.它用于格式化此函数结果的代码可以在 https://github.com/sympy/sympy_gamma/blob/master/app/logic/intsteps.py.

除非您使用 SymPy Gamma，否则 integral_steps 的输出将难以阅读.示例:

Unless you are using SymPy Gamma, the output of integral_steps will be hard to read. Example:

from sympy.integrals.manualintegrate import integral_steps
integral_steps(x*sin(3*x), x)

返回

PartsRule(u=x, dv=sin(3*x), v_step=URule(u_var=_u, u_func=3*x, constant=1/3, substep=ConstantTimesRule(constant=1/3, other=sin(_u), substep=TrigRule(func='sin', arg=_u, context=sin(_u), symbol=_u), context=sin(_u), symbol=_u), context=sin(3*x), symbol=x), second_step=ConstantTimesRule(constant=-1/3, other=cos(3*x), substep=URule(u_var=_u, u_func=3*x, constant=1/3, substep=ConstantTimesRule(constant=1/3, other=cos(_u), substep=TrigRule(func='cos', arg=_u, context=cos(_u), symbol=_u), context=cos(_u), symbol=_u), context=cos(3*x), symbol=x), context=-cos(3*x)/3, symbol=x), context=x*sin(3*x), symbol=x)

SymPy Gamma 网站.

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