On some generalization of Frullani and Dirichlet formulas. (Ukrainian) Zbl 0924.26002

Kalajda, O. F.; Melnychuk, D. V.

Visn., Ser. Fiz.-Mat. Nauky, Kyïv. Univ. Im. Tarasa Shevchenka 1997, No. 1, 57-59 (1997).

The authors give formulas for the calculation of the following integrals depending on parameters: \(I_{u}(\alpha,\beta)=\int_{a}^{b}[f(\alpha u(x))-f(\beta u(x))]/v(x)dx\) and \(D_{u}(\beta)=\int\limits_{a}^{b}\sin(\beta u(x))/v(x)dx\), where \(f\) is a continuous function; \(u\) is a monotonic continuously-differentiable function; \(u(a)=0, \lim_{x\to b}u(x)=+\infty, v(x)u'(x)=\gamma u(x), \gamma=\text{const}\).

Reviewer: A.D.Borisenko (Kyïv)

MSC:

26A06	One-variable calculus
26A48	Monotonic functions, generalizations

Keywords:

generalization; Frullani formula; Dirichlet formula