Integration with binomial form from Gradshteyn

Question

${\int x^n z^m_k dx} ={x^{n+1}z^m_k \over {km+n+1}}+{{amk \over km+n+1} \int x^n z^{m-1}_k }dx $

where $z^m_k = (a+bx^k)$

How can I solve this kind of problem? need some hint.

Answer 1

Integration by parts gives \begin{eqnarray*} \int x^n(a+bx^k)^m dx =\frac{x^{n+1}}{n+1} (a+bx^k)^m -\frac{kmb}{n+1} \int x^{n+k}(a+bx^k)^{m-1} dx. \end{eqnarray*} Note that the second intergral can be rewritten as \begin{eqnarray*} b\int x^{n+k}(a+bx^k)^{m-1} dx= \int x^{n}(-a+a+bx^k)(a+bx^k)^{m-1} dx \\= -a \int x^{n}(a+bx^k)^{m-1} dx +\int x^n(a+bx^k)^m dx. \end{eqnarray*} The second integral is the integral on the LHS of the first equation; the result now follows using a little bit of linear algebra.