Lobachevsky integral formula explained

In mathematics, Dirichlet integrals play an important role in distribution theory. We can see the Dirichlet integral in terms of distributions.

One of those is the improper integral of the sinc function over the positive real line,

	infty
\int
	0

	\sinx
	x

	infty
=\int
	0

	\sin²x
	x²

dx=

	\pi
	2.

Lobachevsky's Dirichlet integral formula

Let

f(x)

be a continuous function satisfying the

\pi

-periodic assumption

f(x+\pi)=f(x)

, and

f(\pi-x)=f(x)

, for

0\leqx<infty

. If the integral

	infty
\int
	0

	\sinx
	x

f(x)dx

is taken to be an improper Riemann integral, we have Lobachevsky's Dirichlet integral formula

	infty
\int
	0

	\sin²x
	x²

f(x)dx=

infty	\sinx
	x

\int

f(x)dx=

	\pi/2
\int
	0

f(x)dx

Moreover, we have the following identity as an extension of the Lobachevsky Dirichlet integral formula^[1]

	infty
\int
	0

	\sin^4x
	x⁴

f(x)dx=

	\pi/2
\int
	0

f(t)dt-

	2
	3

	\pi/2
\int
	0

\sin^2tf(t)dt.

As an application, take

f(x)=1

. Then

	infty
\int
	0

	\sin^4x
	x⁴

dx=

	\pi
	3

References

G. H.. Hardy. The Integral
infty
\int
0
\sinx
x

dx=

\pi
2,

. The Mathematical Gazette. 5. 80. 1909. 98–103. 10.2307/3602798 . 3602798.
Alfred Cardew . Dixon. Proof That
infty
\int
0
\sinx
x

dx=

\pi
2,

. The Mathematical Gazette. 6. 96 . 1912. 223–224. 10.2307/3604314 . 3604314.

Notes and References

Jolany. Hassan. An extension of Lobachevsky formula . . 73 . 3 . 2018 . 89–94 . 10.4171/EM/358. 1004.2653 .