MHB Definite integral challenge....

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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.
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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) ]
 
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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.
 
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