INTEGRALS CONTAINING ax+b AND px+q

1.

$\displaystyle \int\displaystyle \frac{dx}{(ax+b)(px+q)}=\displaystyle \frac{1}{bp-aq}\ln\left(\displaystyle \frac{px+q}{ax+b}\right)$

2.

$\displaystyle \int\displaystyle \frac{x\,dx}{(ax+b)(px+q)}=\displaystyle \frac{... ...\{\displaystyle \frac{b}{a}\ln(ax+b)-\displaystyle \frac{q}{p}\ln(px+q)\right\}$

3.

$\displaystyle \int\displaystyle \frac{dx}{(ax+b)^{\displaystyle 2}(px+q)}=\disp... ...laystyle \frac{p}{bp-aq}\ln\left(\displaystyle \frac{px+q}{ax+b}\right)\right\}$

4.

$\displaystyle \int\displaystyle \frac{x\,dx}{(ax+b)^{\displaystyle2}(px+q)}=\di... ...(\displaystyle \frac{ax+b}{px+q}\right)-\displaystyle \frac{b}{a(ax+b)}\right\}$

5.

$\displaystyle\begin{array}{l} \displaystyle \int\displaystyle \frac{x^{\display... ...\displaystyle \frac{b(bp-2aq)}{a^{\displaystyle2}}\ln(ax+b)\right\} \end{array}$

6.

$\displaystyle\begin{array}{l} \displaystyle \int\displaystyle \frac{dx}{(ax+b)^... ...rac{dx}{(ax+b)^{\displaystyle m}(px+q)^{\displaystyle n-1}}\right\} \end{array}$

7.

$\displaystyle \int\displaystyle \frac{ax+b}{px+q}\,dx=\displaystyle \frac{ax}{p}+\displaystyle \frac{bp-aq}{p^{\displaystyle2}}\ln(px+q)$

8.

$\displaystyle \int\displaystyle \frac{(ax+b)^{\displaystyle m}}{(px+q)^{\displa... ...displaystyle m-1}}{(px+q)^{\displaystyle n-1}}\,dx\right\} \end{array} \right.$