# Integrals are equal? How?

Why are these integrals the same?

$$\int_0^1\bigg(\frac{-1}{\ln^3(x)}\bigg)\exp\bigg(\frac{1}{\ln(x)}\bigg)~dx=\int_0^1 \bigg(-\ln(x)\bigg)\exp\bigg(\frac{1}{\ln(x)}\bigg)~dx=2K_2(2)$$

Where the $$K_2$$ is a modified Bessel function.

Is there some property that describes when this happens?

I think it has to do with the functions being in the same general form, but I can't pinpoint the specifics.

Thanks.

• This is quite nice and it's not hard to show that they are equal (by easy, I mean elementary without using some special functions). How did you find those integrals? Commented Oct 31, 2019 at 23:19
• @Nyssa I knew that $\int_0^1 \exp(\frac{1}{\ln(x)})dx=2K_1(2)$ and I wanted to find similar integrals for $2K_n(2)$ $n=0,2,3,...$ Then I noticed that pairs of integrals had the same area. Commented Oct 31, 2019 at 23:27
• @Zacky. This integrals are not equal.For first: $2 K_2(2)$ and second:$2 K_0(2)$ ?For general: $$\int_0^1 -\frac{\exp \left(\frac{1}{\ln (x)}\right)}{\ln ^n(x)} \, dx=-2 (-1)^{-n} K_{-1+n}(2)$$ Commented Apr 24, 2020 at 15:41

We can show that the difference between them is $$0$$ which shows how they are equal. $$\int_0^1 \exp\left(\frac{1}{\ln x}\right)\left(\ln x-\frac{1}{\ln^3 x}\right)dx\overset{\ln x \to x}=\int_{-\infty}^0 e^{x+1/x}\left(x-\frac{1}{x^3}\right)dx$$ $$=\int_{-\infty}^0 e^{x+1/x}\left(1-\frac{1}{x^2}\right)\left(x+\frac{1}{x}\right)dx=\int_{-\infty}^0 \left(e^{x+1/x}\right)'\left(x+\frac{1}{x}\right)dx$$ $$\overset{IBP}=-\int_{-\infty}^0 e^{x+1/x}\left(1-\frac{1}{x^2}\right)dx=-e^{x+1/x}\bigg|_{-\infty}^0=0$$
This can be further generalized. A more direct approach than the one from above is as follows: $$I(n)=\int_0^1 \exp\left(\frac{1}{\ln x}\right)\left(\ln^{n-1} x-\frac{1}{\ln^{n+1}x}\right)dx\overset{\ln x\to x}=\int_{-\infty}^0 e^{x+1/x}\left(x^{n-1}-\frac{1}{x^{n+1}}\right)dx$$ $$\overset{x\to 1/x}=\int_{-\infty}^0 e^{x+1/x}\left(\frac{1}{x^{n+1}}-x^{n-1}\right)dx\Rightarrow 2I(n)=0$$
$$\Rightarrow \int_0^1 \exp\left(\frac{1}{\ln x}\right)\ln^{n-1} xdx=\int_0^1 \exp\left(\frac{1}{\ln x}\right)\frac{1}{\ln^{n+1}x} dx$$