Another integral for Lobachvesky
A miserious integral related to Lobachevsky
Today we show the proof of this strange integral posted by
@integralsbot
∫∞0sin(tan(x))xdx=π2(1−1e)
The trick to solve this integral is a variation to the
Lobachevsky integral formula for odd functions and the application of the
residue theorem.
Proof:
Recall the
Lobachevsky integral formula for odd functions:
Theorem. Let
f(x) an odd, complex-valued,
π-periodic function such that
f(x)tanx is absolutely integrable over a single period.
Therefore
∫∞0f(x)xdx=∫π20f(x)tan(x)dx
Back to the integral:
Note that the function
f(x)=sin(tan(x))
is an odd function that satisfies all the hypothesis of the theorem.
Hence
I=∫∞0sin(tan(x))xdx=∫π20sin(tan(x))tan(x)dx from (*) =∫π20sin(w)w(1+w2)dw(w↦tanx)=∫∞0sin(w)wdw−∫∞0wsin(w)1+w2dw=∫∞0sin(w)wdw−12∫∞−∞wsin(w)1+w2dw(the integrand is even)=∫∞0sin(w)wdw⏟Dirichlet integral−ℑ(∫∞−∞weiw1+w2dw⏟J)=π2−ℑ(∫∞−∞weiw1+w2dw⏟J)
The first integral is the classic
Dirichlet integral, the second is a little bit more difficult:
Let
R>1 and consider the following complex function
f(z)=zeiz1+z2
This function has two poles at
z=±i.
Now consider the following contour (half upper circle):
C1=tt∈(−R,R)
C2=Reiθθ∈[0,π]
Hence
∮C=C1∪C2zeiz1+z2dz=∫C1f(z)dz+∮C2f(dz)dz=∫R−Rteit1+t2dt+∫π0iR2e2iθeiReiθ1+Reiθdθ
We will show that the integral in the right hand side will vanish as
R→∞. For that end we will make use of the
Jordan's Lemma:
Jordan's lemma. Consider a complex-valued, continuous function
f, defined on a semicircular contour
CR={Reiθ∣θ∈[0,π]}
of positive radius
R lying in the upper half-plane, centered at the origin. If the function
f is of the form
f(z)=eiazg(z),z∈CR,
with a positive parameter
a, then Jordan's lemma states the following upper bound for the contour integral:
|∫CRf(z)dz|≤πaMRwhereMR:=maxθ∈[0,π]|g(Reiθ)|
with equality when
g vanishes everywhere, in which case both sides are identically zero.
Applying the lemma to
f(z):
f(z)=zeiz1+z2=eizg(z) where g(z)=z1+z2
Hence
|∫C2zeiz1+z2dz|≤πMR
where
MR=maxθ∈[0,π]|g(Reiθ)|=maxθ∈[0,π]|Reiθ1+R2ei2θ|=maxθ∈[0,π]R|1+R2ei2θ|=maxθ∈[0,π]R|1−(−R2ei2θ)|≤R|1−R2|=RR2−1(R>1)
Therefore
|∫C2zeiz1+z2dz|≤πMR≤πRR2−1
Taking the limit as
R→∞
∫C2zeiz1+z2dz=∫π0iR2e2iθeiReiθ1+Reiθdθ=0
Hence, since the only pole in the upper half circle is
z=i we have by the residue theorem
J=∫∞−∞teit1+t2dt=∮Czeiz1+z2dz=2πiRes(f,i)=limz→i2πizeiz(z+i)=πie
Therefore
I=π2−ℑ(J)=π2−12ℑ(πie)=π2−π2e=π2(1−1e)
Hence, we can conclude
∫∞0sin(tan(x))xdx=π2(1−1e)