Upon introducing new integration variable z=x-b (or w=x-a) your integral can be expressed via the so-called incomplete beta function, see e.g. http://en.wikipedia.org/wiki/Beta_function#Incomplete_beta_function .
Dear Prof. Sergyeyev
Thanks a lot for your answer. But the problem is not solved by this way.
Best regards
Numerical quadrature will give you inexact value. The error will be especially large when b is such that b+x=0 somewhere in [0,1]. For n fixed (integer) you will get your precise answer after asking your favorite symbolic math package. For n=1,2,3 and 4 the result is a sum of polynomials in a and b multiplied by expressions like log(b) and log(b+1) what suggests that maxima proceeds integrating per parts after asking whether b+1 is positive, negative or 0).
Since by the binomial theorem (C(n,k) are binomial coefficients)
((a+x)/(b+x))^n=(1+(a-b)/(b+x))^n=∑_(k=0)^nC(n,k) (a-b)^k/(b+x)^k
the integral in the question is reduced to the simple one
∫(b+x)^(-k) dx
Oh thank you very much Gro and Marek for your cooperation.
Best regards
This problem has an exact answer found after setting z=b+x, c=a-b and using the binomial expansion. Also it is one which meets the requirement that the integral equals x when c=0. Will type put the answer here plus give a jpg of the solution below-
Int{(a+x)/(b+x),x}=x+ncln(b+x)-Sum{C[n,k]c^k/[(k-1)*(b+x)^(k-1)],k=2..n}
where C[n,k]is the binomial coefficient.
This problem has an exact answer found after setting z=b+x, c=a-b and using the binomial expansion. Also it is one which meets the requirement that the integral equals x when c=0. Will type the answer here plus give a jpg of the solution below-
J(a,b,n,x)=Int{(a+x)/(b+x)^n,x}=x+ncln(b+x)-Sum{C[n,k]c^k/[(k-1)*(b+x)^(k-1)],k=2..n}
where C[n,k]is the binomial coefficient.
My answer is done by http://www.wolframalpha.com/ about indefinite integral and it is shown on the following figure.
My answer for the integral definite in the cases n in N is done on the fiugure bellow. Your sincerely: Anna Tomova.
My answer for the integral indefinite and definite in the cases n in Q is done by http://www.wolframalpha.com/ on the figures bellow . Your sincerely: Anna Tomova.
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