Integration of sin(ax^n), cos(ax^n), ect.
Chan Ching Chuen
: I came upon the following difficult integration problem, and I hope some-one
: with a lot of patience and will power [or MATHEMATICA :-)] could help me:
: What is the integral of sin(ax+bx^2), sin(ax+bx^2+cx^3), sin(ax+bx^2+cx^
: 3), ...
: What is the corresponding integrals for cos(...) ?
: Thanx
: Jaco Maritz
For sin(ax+bx^2), Maple gives
{sqrt(2pi)/2sqrt(b)}{cos(a^2/4b)FS(sqrt(2/(b*pi))(bx+a/2))
-sin(a^2/4b)FC(sqrt(2/(b*pi))(bx+a/2))}
where
FS(y)=int(sin(pi/2*t^2), t=0..x)
FC(y)=int(cos(pi/2*t^2), t=0..y)
C C Chan
Laurie, DP
>I came upon the following difficult integration problem, and I hope some-one
>with a lot of patience and will power [or MATHEMATICA :-)] could help me:
>What is the integral of sin(ax+bx^2), sin(ax+bx^2+cx^3), sin(ax+bx^2+cx^
>3), ...
>What is the corresponding integrals for cos(...) ?
>Thanx
>Jaco Maritz
Can't be done in terms of elementary functions. Even the simplest of
your cases i.e. sin(x^2) requires the Fresnel functions.
Gradshteyn and Ryzhik (Tables of integrals ..., Academic Press, 1980) give
the integral of sin(ax^2+bx+c) (and cos) in terms of the Fresnel functions.
iiiiiii Dirk Laurie, Department of Mathematics and Computer Science
\ I / Potchefstroom University for Christian Higher Education
I P.O. Box 1174, Vanderbijlpark 1900, South Africa
In Thy Light wsk...@puknet.puk.ac.za Tel:(27)(16)807-3600 Fax:...-3614
Kevin M. Johnson
: I came upon the following difficult integration problem, and I hope some-one
: What is the integral of sin(ax+bx^2), sin(ax+bx^2+cx^3), sin(ax+bx^2+cx^
: 3), ...
: What is the corresponding integrals for cos(...) ?
: Thanx
: Jaco Maritz
n[15]:=
Integrate[Sin[A*x+B],x]
Out[15]=
Cos[B + A x]
-(------------)
A
In[17]:=
Integrate[Cos[A*x+B],x]
Out[17]=
Sin[B + A x]
------------
A
In[16]:=
Integrate[Sin[A*x^2+B*x+C],x]
Out[16]=
2
Pi B B + 2 A x
(Sqrt[--] (Cos[--- - C] FresnelS[------------------] -
2 4 A Sqrt[A] Sqrt[2 Pi]
2
B + 2 A x B
FresnelC[------------------] Sin[--- - C])) / Sqrt[A]
Sqrt[A] Sqrt[2 Pi] 4 A
In[18]:=
Integrate[Cos[A*x^2+B*x+C],x]
Out[18]=
2
Pi B B + 2 A x
(Sqrt[--] (Cos[--- - C] FresnelC[------------------] +
2 4 A Sqrt[A] Sqrt[2 Pi]
2
B + 2 A x B
FresnelS[------------------] Sin[--- - C])) / Sqrt[A]
Sqrt[A] Sqrt[2 Pi] 4 A
Not sure how much this helps you though....
--
Kevin Johnson
k...@bilbo.baylor.edu
"There are two kinds of geniuses. An ordinary genius is a fellow that
you and I would be just as good as, if we were only many times better.
The other kind are magicians. Even after we understand what they have
done, the process by which they have done it is completely dark."
-- Mark Kac, Mathematician
Terry Greeniaus
> In article <jmaritz.8...@firga.sun.ac.za> jma...@firga.sun.ac.za (Jaco
>
> >I came upon the following difficult integration problem, and I hope some-one
> >with a lot of patience and will power [or MATHEMATICA :-)] could help me:
>
> >What is the integral of sin(ax+bx^2), sin(ax+bx^2+cx^3), sin(ax+bx^2+cx^
> >3), ...
> >What is the corresponding integrals for cos(...) ?
>
> >Thanx
>
> >Jaco Maritz
>
> Can't be done in terms of elementary functions. Even the simplest of
> your cases i.e. sin(x^2) requires the Fresnel functions.
>
> Gradshteyn and Ryzhik (Tables of integrals ..., Academic Press, 1980) give
> the integral of sin(ax^2+bx+c) (and cos) in terms of the Fresnel functions.
Here's what Mathematica gave me:
Integrate[Sin(a*x+b*x^2),x]
2 3
Sin a x Sin b x
-------- + --------
2 3
Integrate[Sin(a*x+b*x^2+c*x^3),x]
2 3 4
Sin a x Sin b x Sin c x
-------- + -------- + --------
2 3 4
Integrate[Sin(a*x+b*x^2+c*x^3+d*x^4),x]
2 3 4 5
Sin a x Sin b x Sin c x Sin d x
-------- + -------- + -------- + --------
2 3 4 5
Integrate[Cos(a*x+b*x^2),x]
2 3
Cos a x Cos b x
-------- + --------
2 3
Integrate[Cos(a*x+b*x^2+c*x^3),x]
2 3 4
Cos a x Cos b x Cos c x
-------- + -------- + --------
2 3 4
Integrate[Cos(a*x+b*x^2+c*x^3+d*x^4),x]
2 3 4 5
Cos a x Cos b x Cos c x Cos d x
-------- + -------- + -------- + --------
2 3 4 5
The pattern seems pretty obvious to me. Mathematica had no trouble
computing these numbers for me. (Hope I never made a mistake!)
--
Terry Greeniaus tgr...@ersys.edmonton.ab.ca
Edmonton Remote Systems Serving Edmonton/Northern Alberta since 1982
Kevin M. Johnson
: wsk...@puknet.puk.ac.za (Laurie, DP) writes:
: Integrate[Sin(a*x+b*x^2),x]
: Integrate[Sin(a*x+b*x^2+c*x^3),x]
: Integrate[Sin(a*x+b*x^2+c*x^3+d*x^4),x]
: Integrate[Cos(a*x+b*x^2),x]
: Integrate[Cos(a*x+b*x^2+c*x^3),x]
: Integrate[Cos(a*x+b*x^2+c*x^3+d*x^4),x]
To use functions on mathematica, you must use brackets [] at all time.
In the above post, mathematica evaluated Sin and Cos as variables.
Try Integrate[Cos[x^2],x] , etc.