def print_CyclicParts(self, rule):
with self.new_step():
self.append("Use integration by parts, noting that the integrand" " eventually repeats itself.")
u, v, du, dv = map(lambda f: sympy.Function(f)(rule.symbol), "u v du dv".split())
current_integrand = rule.context
total_result = sympy.S.Zero
with self.new_level():
sign = 1
for rl in rule.parts_rules:
with self.new_step():
self.append("For the integrand {}:".format(self.format_math(current_integrand)))
self.append(
"Let {} and let {}.".format(
self.format_math(sympy.Eq(u, rl.u)), self.format_math(sympy.Eq(dv, rl.dv))
)
)
v_f, du_f = _manualintegrate(rl.v_step), rl.u.diff(rule.symbol)
total_result += sign * rl.u * v_f
current_integrand = v_f * du_f
self.append(
"Then {}.".format(
self.format_math(
sympy.Eq(
sympy.Integral(rule.context, rule.symbol),
total_result - sign * sympy.Integral(current_integrand, rule.symbol),
)
)
)
)
sign *= -1
with self.new_step():
self.append("Notice that the integrand has repeated itself, so " "move it to one side:")
self.append(
"{}".format(
self.format_math_display(
sympy.Eq(
(1 - rule.coefficient) * sympy.Integral(rule.context, rule.symbol), total_result
)
)
)
)
self.append("Therefore,")
self.append(
"{}".format(
self.format_math_display(
sympy.Eq(sympy.Integral(rule.context, rule.symbol), _manualintegrate(rule))
)
)
)
def print_CyclicParts(self, rule):
with self.new_step():
self.append("Use integration by parts, noting that the integrand"
" eventually repeats itself.")
u, v, du, dv = [
sympy.Function(f)(rule.symbol) for f in 'u v du dv'.split()
]
current_integrand = rule.context
total_result = sympy.S.Zero
with self.new_level():
sign = 1
for rl in rule.parts_rules:
with self.new_step():
self.append("For the integrand {}:".format(
self.format_math(current_integrand)))
self.append("Let {} and let {}.".format(
self.format_math(sympy.Eq(u, rl.u)),
self.format_math(sympy.Eq(dv, rl.dv))))
v_f, du_f = _manualintegrate(rl.v_step), rl.u.diff(
rule.symbol)
total_result += sign * rl.u * v_f
current_integrand = v_f * du_f
self.append("Then {}.".format(
self.format_math(
sympy.Eq(
sympy.Integral(rule.context, rule.symbol),
total_result - sign * sympy.Integral(
current_integrand, rule.symbol)))))
sign *= -1
with self.new_step():
self.append(
"Notice that the integrand has repeated itself, so "
"move it to one side:")
self.append("{}".format(
self.format_math_display(
sympy.Eq((1 - rule.coefficient) *
sympy.Integral(rule.context, rule.symbol),
total_result))))
self.append("Therefore,")
self.append("{}".format(
self.format_math_display(
sympy.Eq(sympy.Integral(rule.context, rule.symbol),
_manualintegrate(rule)))))
def print_Log(self, rule):
with self.new_step():
self.append(
"The integral of {} is {}.".format(
self.format_math(1 / rule.func), self.format_math(_manualintegrate(rule))
)
)
def print_Add(self, rule):
with self.new_step():
self.append("Integrate term-by-term:")
for substep in rule.substeps:
with self.new_level():
self.print_rule(substep)
self.append("The result is: {}".format(self.format_math(_manualintegrate(rule))))
def print_U(self, rule):
with self.new_step(), self.new_u_vars() as (u, du):
# commutative always puts the symbol at the end when printed
dx = sympy.Symbol('d' + rule.symbol.name, commutative=0)
self.append("Let {}.".format(
self.format_math(sympy.Eq(u, rule.u_func))))
self.append("Then let {} and substitute {}:".format(
self.format_math(
sympy.Eq(du,
rule.u_func.diff(rule.symbol) * dx)),
self.format_math(rule.constant * du)))
integrand = rule.constant * \
rule.substep.context.subs(rule.u_var, u)
self.append(self.format_math_display(sympy.Integral(integrand, u)))
with self.new_level():
self.print_rule(
replace_u_var(rule.substep, rule.symbol.name, u))
self.append(
'<div class="collapsible"><h2>open answer</h2><ol class="content">Now replace {} to get:'
.format(self.format_math(u)))
self.append(
self.format_math_display(_manualintegrate(rule)) +
'</ol></div>')
def print_Arctan(self, rule):
with self.new_step():
self.append(
"The integral of {} is {}.".format(
self.format_math(1 / (1 + rule.symbol ** 2)), self.format_math(_manualintegrate(rule))
)
)
def print_Constant(self, rule):
with self.new_step():
self.append("The integral of a constant is the constant "
"times the variable of integration:")
self.append(
self.format_math_display(
sympy.Eq(sympy.Integral(rule.constant, rule.symbol),
_manualintegrate(rule))))
def print_Constant(self, rule):
with self.new_step():
self.append("The integral of a constant is the constant "
"times the variable of integration:")
self.append(
self.format_math_display(
sympy.Eq(sympy.Integral(rule.constant, rule.symbol),
_manualintegrate(rule))))
def print_Add(self, rule):
with self.new_step():
self.append("Integrate term-by-term:")
for substep in rule.substeps:
with self.new_level():
self.print_rule(substep)
self.append("The result is: {}".format(
self.format_math(_manualintegrate(rule))))
def print_Exp(self, rule):
with self.new_step():
if rule.base == sympy.E:
self.append("The integral of the exponential function is itself.")
else:
self.append("The integral of an exponential function is itself"
" divided by the natural logarithm of the base.")
self.append(self.format_math_display(
sympy.Eq(sympy.Integral(rule.context, rule.symbol),
_manualintegrate(rule))))
def print_Exp(self, rule):
with self.new_step():
if rule.base == sympy.E:
self.append("The integral of the exponential function is itself.")
else:
self.append("The integral of an exponential function is itself"
" divided by the natural logarithm of the base.")
self.append(self.format_math_display(
sympy.Eq(sympy.Integral(rule.context, rule.symbol),
_manualintegrate(rule))))
def print_Add(self, rule):
with self.new_step():
self.append("Integrate term-by-term:")
for substep in rule.substeps:
with self.new_level():
self.print_rule(substep)
self.append(
'<div class="collapsible"><h2>open answer</h2><ol class="content">The result is: '
)
self.append(
self.format_math(_manualintegrate(rule)) + '</ol></div>')
def print_Power(self, rule):
with self.new_step():
self.append("The integral of {} is {} when {}:".format(
self.format_math(rule.symbol ** sympy.Symbol('n')),
self.format_math((rule.symbol ** (1 + sympy.Symbol('n'))) /
(1 + sympy.Symbol('n'))),
self.format_math(sympy.Ne(sympy.Symbol('n'), -1)),
))
self.append(
self.format_math_display(
sympy.Eq(sympy.Integral(rule.context, rule.symbol),
_manualintegrate(rule))))
def print_Power(self, rule):
with self.new_step():
self.append("The integral of {} is {} when {}:".format(
self.format_math(rule.symbol**sympy.Symbol('n')),
self.format_math((rule.symbol**(1 + sympy.Symbol('n'))) /
(1 + sympy.Symbol('n'))),
self.format_math(sympy.Ne(sympy.Symbol('n'), -1)),
))
self.append(
self.format_math_display(
sympy.Eq(sympy.Integral(rule.context, rule.symbol),
_manualintegrate(rule))))
def print_ConstantTimes(self, rule):
with self.new_step():
self.append("The integral of a constant times a function "
"is the constant times the integral of the function:")
self.append(self.format_math_display(
sympy.Eq(
sympy.Integral(rule.context, rule.symbol),
rule.constant * sympy.Integral(rule.other, rule.symbol))))
with self.new_level():
self.print_rule(rule.substep)
self.append("So, the result is: {}".format(
self.format_math(_manualintegrate(rule))))
def print_ConstantTimes(self, rule):
with self.new_step():
self.append("The integral of a constant times a function "
"is the constant times the integral of the function:")
self.append(self.format_math_display(
sympy.Eq(
sympy.Integral(rule.context, rule.symbol),
rule.constant * sympy.Integral(rule.other, rule.symbol))))
with self.new_level():
self.print_rule(rule.substep)
self.append("So, the result is: {}".format(
self.format_math(_manualintegrate(rule))))
def print_Trig(self, rule):
with self.new_step():
text = {
'sin': "The integral of sine is negative cosine:",
'cos': "The integral of cosine is sine:",
'sec*tan': "The integral of secant times tangent is secant:",
'csc*cot': "The integral of cosecant times cotangent is cosecant:",
}.get(rule.func)
if text:
self.append(text)
self.append(self.format_math_display(
sympy.Eq(sympy.Integral(rule.context, rule.symbol),
_manualintegrate(rule))))
def print_Trig(self, rule):
with self.new_step():
text = {
'sin': "The integral of sine is negative cosine:",
'cos': "The integral of cosine is sine:",
'sec*tan': "The integral of secant times tangent is secant:",
'csc*cot': "The integral of cosecant times cotangent is cosecant:",
}.get(rule.func)
if text:
self.append(text)
self.append(self.format_math_display(
sympy.Eq(sympy.Integral(rule.context, rule.symbol),
_manualintegrate(rule))))
def finalize(self):
rule = filter_unknown_alternatives(self.rule)
answer = _manualintegrate(rule)
if answer:
simp = sympy.simplify(sympy.trigsimp(answer))
if simp != answer:
answer = simp
with self.new_step():
self.append("Now simplify:")
self.append(self.format_math_display(simp))
with self.new_step():
self.append("Add the constant of integration:")
self.append(self.format_math_constant(answer))
self.lines.append('</ol>')
self.lines.append('<hr/>')
self.level = 0
self.append('The answer is:')
self.append(self.format_math_constant(answer))
return '\n'.join(self.lines)
def finalize(self):
rule = filter_unknown_alternatives(self.rule)
answer = _manualintegrate(rule)
if answer:
simp = sympy.simplify(sympy.trigsimp(answer))
if simp != answer:
answer = simp
with self.new_step():
self.append("Now simplify:")
self.append(self.format_math_display(simp))
with self.new_step():
self.append("Add the constant of integration:")
self.append(self.format_math_constant(answer))
self.lines.append('</ol>')
self.lines.append('<hr/>')
self.level = 0
self.append('The answer is:')
self.append(self.format_math_constant(answer))
return '\n'.join(self.lines)
def print_ConstantTimes(self, rule):
with self.new_step():
self.append(
"The integral of <strong>a</strong> constant times a function "
"is <strong>the</strong> constant times the integral of the function:"
)
self.append(
self.format_math_display(
sympy.Eq(
sympy.Integral(rule.context, rule.symbol),
rule.constant *
sympy.Integral(rule.other, rule.symbol))))
with self.new_level():
self.print_rule(rule.substep)
self.append(
'<div class="collapsible"><h2>open answer</h2><ol class="content">So, the result is: '
)
self.append(
self.format_math(_manualintegrate(rule)) + '</ol></div>')
def print_U(self, rule):
with self.new_step(), self.new_u_vars() as (u, du):
# commutative always puts the symbol at the end when printed
dx = sympy.Symbol("d" + rule.symbol.name, commutative=0)
self.append("Let {}.".format(self.format_math(sympy.Eq(u, rule.u_func))))
self.append(
"Then let {} and substitute {}:".format(
self.format_math(sympy.Eq(du, rule.u_func.diff(rule.symbol) * dx)),
self.format_math(rule.constant * du),
)
)
integrand = rule.substep.context.subs(rule.u_var, u)
self.append(self.format_math_display(sympy.Integral(integrand, u)))
with self.new_level():
self.print_rule(replace_u_var(rule.substep, rule.u_var, u))
self.append("Now substitute {} back in:".format(self.format_math(u)))
self.append(self.format_math_display(_manualintegrate(rule)))
def finalize(self):
rule = filter_unknown_alternatives(self.rule)
answer = _manualintegrate(rule)
if answer:
simp = sympy.simplify(sympy.trigsimp(answer))
if simp != answer:
answer = simp
with self.new_step():
self.append(
'<div class="collapsible"><h2>open answer</h2><ol class="content">Now simplify:'
)
self.append(self.format_math_display(simp) + '</ol></div>')
with self.new_step():
self.append(
'<div class="collapsible"><h2>open answer</h2><ol class="content">Add the constant of integration to get:'
)
self.append(self.format_math_constant(answer) + '</ol></div>')
self.lines.append('</ol>')
self.level = 0
# self.append('The answer is:')
# self.append(self.format_math_constant(answer))
return '\n'.join(self.lines)
def print_U(self, rule):
with self.new_step(), self.new_u_vars() as (u, du):
# commutative always puts the symbol at the end when printed
dx = sympy.Symbol('d' + rule.symbol.name, commutative=0)
self.append("Let {}.".format(
self.format_math(sympy.Eq(u, rule.u_func))))
self.append("Then let {} and substitute {}:".format(
self.format_math(
sympy.Eq(du,
rule.u_func.diff(rule.symbol) * dx)),
self.format_math(rule.constant * du)))
integrand = rule.substep.context.subs(rule.u_var, u)
self.append(self.format_math_display(sympy.Integral(integrand, u)))
with self.new_level():
self.print_rule(replace_u_var(rule.substep, rule.u_var, u))
self.append("Now substitute {} back in:".format(
self.format_math(u)))
self.append(self.format_math_display(_manualintegrate(rule)))
def print_Arctan(self, rule):
with self.new_step():
self.append("The integral of {} is {}.".format(
self.format_math(1 / (1 + rule.symbol**2)),
self.format_math(_manualintegrate(rule))))
def print_Log(self, rule):
with self.new_step():
self.append("The integral of {} is {}.".format(
self.format_math(1 / rule.func),
self.format_math(_manualintegrate(rule))))
