Definite integral transform $\int_{x=a}^{x=b}f(x)dx=\int_{t=0}^{exp(-a)}f(-ln(t))\frac{dt}{t}$

Question

I found that transform by Chebyshev:

But it doesn't work with simple function in Mathcad test:

What's wrong with this solution?

Answer 1

To arrive at the integral transformation

$$I=\int_a^b f(x)\,dx=\int_0^\text{exp(-a)}f(-\ln t)\frac{dt}{t}$$

we let $x = -\ln t$ so that $dx=-\dfrac{dt}{t}$ and our integration limits become $$a=-\ln t \implies t=\text{exp(-a)}, \quad b=-\ln t \implies t=\text{exp(-b)}$$

therefore

$$I=\int_\text{exp(-a)}^\text{exp(-b)}f(-\ln t)\left(-\frac{dt}{t}\right)=\int_\text{exp(-b)}^\text{exp(-a)}f(-\ln t)\frac{dt}{t}$$

where the lower limit goes to zero as $b\to \infty$. In your test, $b$ is a small finite number. The original author must have deduced that $b\to\infty$ before writing equation $(30)$.