Integrate a^(px)b^(mx)dx

Question

Integrate with whole solution

Answer 1

Another method:

 

Let A=ln(a) and B=ln(b)

 

a^px = e^Apx

b^mx = e^Bmx

 

a^pxb^mx = e^Apxe^Bmx = e^(Ap+Bm)x

 

Let K=(Ap+Bm)

 

∫e^Kxdx = (1/K)e^Kx +C

Answer 2

∫[a^pxb^mx]dx = ∫[(a^p)^x(b^m)^x]dx

 

                   = ∫(a^pb^m)^xdx  Recall:  ∫c^xdx = (1/lnc)c^x + constant

 

                   = 1 / ln(a^pb^m) (a^pb^m)^x + C

 

                   = 1 / ln(a^pb^m) (a^pxb^mx) + C