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$ax+b$ ve $px+q$ içeren integraller

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


3.
$\displaystyle \int\displaystyle \frac{dx}{(ax+b)^{\displaystyle 2}(px+q)}=\displaystyle \frac{1}{bp-aq}\left\{\displaystyle \frac{1}{ax+b}+\displaystyle \frac{p}{bp-aq}\ln\left(\displaystyle \frac{px+q}{ax+b}\right)\right\}$


4.
$\displaystyle\begin{array}{lcl} \displaystyle \int\displaystyle \frac{x\,dx}{(ax+b)^{\displaystyle2}(px+q)}=\displaystyle \frac{1}{bp-aq}\left\{\displaystyle \frac{q}{bp-aq}\ln\left(\displaystyle \frac{ax+b}{px+q}\right)-\displaystyle \frac{b}{a(ax+b)}\right\} \end{array}$


5.
$\displaystyle\begin{array}{lcl} \displaystyle \int\displaystyle \frac{x^{\displaystyle2}\,dx}{(ax+b)^{\displaystyle2}(px+q)}&=&\displaystyle \frac{b^{\displaystyle 2}}{(bp-aq)a^{\displaystyle2}(ax+b)}+\displaystyle \frac{1}{(bp-aq)^{\displaystyle2}}\left\{\displaystyle \frac{q^{\displaystyle2}}{p}\ln(px+q)\right.\\\ && \left. +\displaystyle \frac{b(bp-2aq)}{a^{\displaystyle2}}\ln(ax+b)\right\} \end{array}$


6.
$\displaystyle\begin{array}{lcl} \displaystyle \int\displaystyle \frac{dx}{(ax+b)^{\displaystyle m}(px+q)^{\displaystyle n}}&=&\displaystyle \frac{-1}{(n-1)(bp-aq)} \left\{\displaystyle \frac{1}{(ax+b)^{\displaystyle m-1}(px+q)^{\displaystyle n-1}}\right.\\&&\left.+ a(m+n-2)\displaystyle \int\displaystyle \frac{dx}{(ax+b)^{\displaystyle m}(px+q)^{\displaystyle n-1}}\right\} \end{array}$


7.
$\displaystyle \int\displaystyle \frac{ax+b}{px+q}\,dx=\displaystyle \frac{ax}{p}+\displaystyle \frac{bp-aq}{p^{\displaystyle2}}\ln(px+q)$


8.
$\small \displaystyle \int\displaystyle \frac{(ax+b)^{\displaystyle m}}{(px+q)^{\displaystyle n}}\,dx= \left\{\begin{array}{l} \displaystyle \frac{-1}{(n-1)(bp-aq)}\left\{\displaystyle \frac{(ax+b)^{\displaystyle m+1}}{(px+q)^{\displaystyle n-1}}+(n-m-2)a\displaystyle \int\displaystyle \frac{(ax+b)^{\displaystyle m-1}}{(px+q)^{\displaystyle n-1}}\,dx\right\}\\ \\ \displaystyle \frac{-1}{(n-m-1)p}\left\{\displaystyle \frac{(ax+b)^{\displaystyle m}}{(px+q)^{\displaystyle n-1}}+m(bp-aq)\displaystyle \int\displaystyle \frac{(ax+b)^{\displaystyle m-1}}{(px+q)^{\displaystyle n}}\,dx\right\}\\ \\ \displaystyle \frac{-1}{(n-1)p}\left\{\displaystyle \frac{(ax+b)^{\displaystyle m}}{(px+q)^{\displaystyle n-1}}-ma\displaystyle \int\displaystyle \frac{(ax+b)^{\displaystyle m-1}}{(px+q)^{\displaystyle n-1}}\,dx\right\} \end{array} \right.$


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