This is my work on Chain Integration and Integration by Parts (higher-order integrals). Also included is a differential analysis on cycloids.
See Fresnel integral.
We could approximate the tangent and secant integrals
and
by using the Cauchy Principal Value and integration by parts:
and
with the square Chain Integration formula
where
.
In fact, integrals through CPV are defined when subtracting balanced pole functions - pure powers of
- from original poles, results in functions with remaining possible singularities of size
with
strict).
Notice that both
and
have simple nonzero isolated poles, limiting to scalar multiples of
. Then
and
with
.
Therefore,
is bounded for
and
; as we have the related bounded integral
and we can do scalar multiple comparisons. So by CPV, the above integrals are defined except at isolated poles. Graphs of these integrals for
are found below:
Graphs of
and
Integral of Tan x^2
|
Integral of Sec x^2
|
|
|
[0,15] × [-5, 5]
|
[0,15] × [-5, 5]
|
Finally, we have, approximately:
compared to
for both
and
.
Modulated Integrals
[edit]
We could also approximate the cotangent and cosecant integrals
and
by using an integration by parts, but need to isolate a pole of order 2 at
for each function. We do so by subtracting
from each function to yield bounded functions
at
(in fact, with
for both functions!), applying the same treatment
as previously (it works similarly at all other poles), and then adding the antiderivative
of
back.
Since both integrals have right limit
as
approaches
, we instead add constants so that critical points/inflection points of the cotangent/cosecant integrals, respectively, approach
as
(canonicalization). We thus have:
With graphs seen below:
Graphs of
and
Integral of Cot x^2
|
Integral of csc x^2
|
|
|
[0,15] × [-5, 5]
|
[0,15] × [-5, 5]
|
Higher Order (Stacked) Integrals
[edit]
A general formula for the second-order real integral (second antiderivative) is
Proof
|
by Product Rule and Fundamental Theorem of Calculus I.
|
A general formula for the third-order real integral (third antiderivative) is
Proof
|
by Product Rule and Fundamental Theorem of Calculus I.
|
A general formula for the
th-order real integral (
th antiderivative) is
Proof
|
Assume by induction, that
Then,
by Product Rule, Binomial formula, combination formulas, and Fundamental Theorem of Calculus I, which implies
and we are done. Intuitively, terms in the derivative of the form come from differentiating each integral (but not the power in front of it, thereby obtaining the right side in contributing from each integral term), and cancel out by means of ; but terms of the form are few, with only one contribution from each integral term.
|
So to compute higher-order integrals, no other nonelementary integrals need to be considered except for those possibly equal to
for
(Riemann-integrable functions). To compute from another bound
, it is sufficient for integrals only of the form
to be considered.
Below, the first five antiderivatives of
are computed and graphed, using this method. Since
is undefined at
(in fact,
is an essential singularity, and every antiderivative diverges as
), we start at
instead (a critical minimum point for
, so most suitable):
![{\displaystyle e^{x^{2}+1/x^{2}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/33f28555547f4ddcf7da86f9d481ada02ed0bf39)
![{\displaystyle \int _{1}^{x}e^{t^{2}+1/t^{2}}~{\text{d}}t}](https://wikimedia.org/api/rest_v1/media/math/render/svg/853916681adddbf5a9ae603937c44a75bc778a59)
![{\displaystyle \iint _{1}^{x}e^{t^{2}+1/t^{2}}~{\text{d}}t^{2}=(x-1)\int _{1}^{x}e^{t^{2}+1/t^{2}}~{\text{d}}t-\int _{1}^{x}(t-1)e^{t^{2}+1/t^{2}}~{\text{d}}t}](https://wikimedia.org/api/rest_v1/media/math/render/svg/742f278bd32ba56edbe27385823a3651fc97ee52)
![{\displaystyle \iiint _{1}^{x}e^{t^{2}+1/t^{2}}~{\text{d}}t^{3}={\frac {1}{2}}\left[(x-1)^{2}\int _{1}^{x}e^{t^{2}+1/t^{2}}~{\text{d}}t-2(x-1)\int _{1}^{x}(t-1)e^{t^{2}+1/t^{2}}~{\text{d}}t+\int _{1}^{x}(t-1)^{2}e^{t^{2}+1/t^{2}}~{\text{d}}t\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e1e0bce45e1402d7ef06e7d04a8e8e6887a2ccc6)
![{\displaystyle {\underset {4~{\text{times}}}{\int \cdots \int _{1}^{x}}}e^{t^{2}+1/t^{2}}~{\text{d}}t^{4}={\frac {1}{6}}\left[(x-1)^{3}\int _{1}^{x}e^{t^{2}+1/t^{2}}~{\text{d}}t-3(x-1)^{2}\int _{1}^{x}(t-1)e^{t^{2}+1/t^{2}}~{\text{d}}t+3(x-1)\int _{1}^{x}(t-1)^{2}e^{t^{2}+1/t^{2}}~{\text{d}}t-\int _{1}^{x}(t-1)^{3}e^{t^{2}+1/t^{2}}~{\text{d}}t\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f825a93e5ad5d4b01e53711283ff5efe25043e72)
![{\displaystyle {\underset {5~{\text{times}}}{\int \cdots \int _{1}^{x}}}e^{t^{2}+1/t^{2}}~{\text{d}}t^{5}={\frac {1}{24}}\left[(x-1)^{4}\int _{1}^{x}e^{t^{2}+1/t^{2}}~{\text{d}}t-4(x-1)^{3}\int _{1}^{x}(t-1)e^{t^{2}+1/t^{2}}~{\text{d}}t+6(x-1)^{2}\int _{1}^{x}(t-1)^{2}e^{t^{2}+1/t^{2}}~{\text{d}}t-4(x-1)\int _{1}^{x}(t-1)^{3}e^{t^{2}+1/t^{2}}~{\text{d}}t+\int _{1}^{x}(t-1)^{4}e^{t^{2}+1/t^{2}}~{\text{d}}t\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/cf79a6759b175e85f89edbe531e9e2fbcefdd66e)
Note that the general chain integral formula cannot be used since (1)
is not strictly monotone, (2) no balanced essential singularity can be isolated from
, not even
, and (3) even the balanced essential singularity
is not elementary-integrable.
The work-energy formula is intrinsic to any twice continuously differentiable increasing function:
This is by Chain Rule, since
for any twice continuously differentiable increasing function
.
Using the work-energy formula
with
, initial position
, and initial velocity
; we have
by the Fundamental Theorem of Calculus I. So in fact the cycloid is the solution
to Newton's Gravitational Law (where time is measured in
) if a particle is bounded in a heavy point object's gravitational field (negative net energy), with zero angular momentum; if we set
,
, to obtain
.
Other zero-angular-momentum solutions for identical mass include
for zero net energy, and
for positive net energy. For the first case, indeed
by the Power Rule, where the additional constant of
is specific to the cycloid only (which is the total energy); and for the second case, indeed
by the parametric derivative, and the additional constant of
is specific to this second path only (which is the total energy).