My pleasure if u can help me on an interesting integration

[benjaminthelight]

How could i solve the following definite integral of a function f(x)=x^n*(k1+k2*x)^(-n-2)ln(1+x) given that the lower limit is 0 and the upper limit is positive infinity?

\(\displaystyle \int_0^{infty}x^n(k_1+k_2x)^{-n-2}ln(1+x)dx\) for positive constants n, K1 and K2

 

[galactus]

I ran this through Maple because it is too onerous to do by hand and is beyond elementary methods.

Here is what it gave me: The k is \(\displaystyle k_{1}\)

\(\displaystyle \frac{1}{{\Gamma}(n+2)}\left[(\frac{1}{k})^{-n}\cdot k^{-n-2}\left(\frac{-{\pi}csc({\pi}n){\Gamma}(n+2)(\frac{1}{k})^{n}(1-\frac{1}{k})^{-n}}{(-n-1)(\frac{1}{k}-1)}+\frac{(-{\Psi}(-n)+{\pi}cot({\pi}n)-{\gamma}+ln(\frac{1}{k})){\pi}csc({\pi}n)k}{{\Gamma}(-n)k_{2}}\)\(\displaystyle +{\Gamma}(n)\text{hypergeo}([1,1],[1-n],\frac{1}{k}))\right]\)

 

[benjaminthelight]

Thank you very much!

 

[Deleted member 4993]

I ran this through Maple because it is too onerous to do by hand and is beyond elementary methods.

Here is what it gave me: The k is \(\displaystyle k_{1}\) ....what happened to \(\displaystyle k_{2}\) ?!

\(\displaystyle \frac{1}{{\Gamma}(n+2)}\left[(\frac{1}{k})^{-n}\cdot k^{-n-2}\left(\frac{-{\pi}csc({\pi}n){\Gamma}(n+2)(\frac{1}{k})^{n}(1-\frac{1}{k})^{-n}}{(-n-1)(\frac{1}{k}-1)}+\frac{(-{\Psi}(-n)+{\pi}cot({\pi}n)-{\gamma}+ln(\frac{1}{k})){\pi}csc({\pi}n)k}{{\Gamma}(-n)k_{2}}\)\(\displaystyle +{\Gamma}(n)\text{hypergeo}([1,1],[1-n],\frac{1}{k}))\right]\)

 

[galactus]

It is there, in the denominator.

 

[Deleted member 4993]

Found it -- ?(-n)k[sub:2yne5dbc]2[/sub:2yne5dbc]

Boy - it was hiding so well -- thanks

 

[benjaminthelight]

BTW, which Maple version have u used? I tried with Maple 12 but I couldn't get the result rather I got itself.