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$\sqrt{ax+b}$ ve $\sqrt{px+q}$ içeren integraller

1.
$\displaystyle \int\displaystyle \frac{dx}{\displaystyle \sqrt{(ax+b)(px+q)}}=\left\{\begin{array}{l} \displaystyle \frac{2}{\displaystyle \sqrt{ap}}\ln\left(\displaystyle \sqrt{a(px+q)}+\displaystyle \sqrt{p(ax+b)}\right)\\ \\ \displaystyle \frac{2}{\displaystyle \sqrt{-ap}}\tan^{\displaystyle-1}\displaystyle \sqrt{\displaystyle \frac{-p(ax+b)}{a(px+q)}} \end{array} \right.$


2.
$\displaystyle \int\displaystyle \frac{x\,dx}{\displaystyle \sqrt{(ax+b)(px+q)}}=\displaystyle \frac{\displaystyle \sqrt{(ax+b)(px+q)}}{ap}-\displaystyle \frac{bp+aq}{2ap}\displaystyle \int\displaystyle \frac{dx}{\displaystyle \sqrt{(ax+b)(px+q)}}$


3.
$\begin{array}{lcl} \displaystyle \int\displaystyle \sqrt{(ax+b)(px+q)}\,dx&=&\displaystyle \frac{2apx+bp+aq}{4ap}\displaystyle \sqrt{(ax+b)(px+q)}\\ &&\\ && -\displaystyle \frac{(bp-aq)^{\displaystyle2}}{8ap}\displaystyle \int\displaystyle \frac{dx}{\displaystyle \sqrt{(ax+b)(px+q)}}\end{array}$


4.
$\displaystyle \int\displaystyle \sqrt{\displaystyle \frac{px+q}{ax+b}}\,dx=\displaystyle \frac{\displaystyle \sqrt{(ax+b)(px+q)}}{a}+\displaystyle \frac{aq-bp}{2a}\displaystyle \int\displaystyle \frac{dx}{\displaystyle \sqrt{(ax+b)(px+q)}}$


5.
$\displaystyle \int\displaystyle \frac{dx}{(px+q)\displaystyle \sqrt{(ax+b)(px+q)}}=\displaystyle \frac{2\displaystyle \sqrt{ax+b}}{(aq-bp)\displaystyle \sqrt{px+q}}$


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