MHB Definite integral challenge....

AI Thread Summary
The discussion focuses on evaluating the definite integral $$\int_0^z\frac{x^m}{(a+\log x)}\,dx$$ for positive integers $$m$$ and positive real numbers $$a$$ and $$z$$. An antiderivative is found in terms of the exponential integral function, leading to the expression $$e^{-a(m+1)} \text{Ei} \Big( (a+\ln x) (m+1) \Big) + C$$. The conversation notes that additional restrictions on the parameters may be necessary to ensure convergence of the integral. The integration process involves a transformation using the exponential function and highlights the complexity of logarithmic integrals. The thread indicates that more generalized forms will be added to a related topic in the future.
DreamWeaver
Messages
297
Reaction score
0
For $$m \in \mathbb{Z}^+$$, and $$a, \, z \in \mathbb{R} > 0$$, evaluate the definite integral:$$\int_0^z\frac{x^m}{(a+\log x)}\,dx$$[I'll be adding a few generalized forms like this in the logarithmic integrals thread, in Maths Notes, shortly... (Heidy) ]
 
Mathematics news on Phys.org
You can find an antiderivative in terms of the exponential integral.$ \displaystyle \int \frac{x^{m}}{a+\ln x} \ dx = \int \frac{e^{u(m+1)}}{a+u} \ du = e^{-a(m+1)} \int \frac{e^{m(w+1)}}{w} \ dw $

$ \displaystyle = e^{-a(m+1)} \ \text{Ei} \Big( w(m+1) \Big) + C $

$ \displaystyle = e^{-a(m+1)} \ \text{Ei} \Big( (a+u) (m+1) \Big) + C $

$ \displaystyle = e^{-a(m+1)} \ \text{Ei} \Big( (a+\ln x) (m+1) \Big) + C $And I think you need more restrictions on the parameters to guarantee convergence.
 
Seemingly by some mathematical coincidence, a hexagon of sides 2,2,7,7, 11, and 11 can be inscribed in a circle of radius 7. The other day I saw a math problem on line, which they said came from a Polish Olympiad, where you compute the length x of the 3rd side which is the same as the radius, so that the sides of length 2,x, and 11 are inscribed on the arc of a semi-circle. The law of cosines applied twice gives the answer for x of exactly 7, but the arithmetic is so complex that the...
Is it possible to arrange six pencils such that each one touches the other five? If so, how? This is an adaption of a Martin Gardner puzzle only I changed it from cigarettes to pencils and left out the clues because PF folks don’t need clues. From the book “My Best Mathematical and Logic Puzzles”. Dover, 1994.
Back
Top