Question

Интегрирование рациональных выражений. $\int\limits^a_b {\frac{dx}{2x^{2}-4x+5 } }$

Answer 1

$\int_{b}^{a}\frac{dx}{2x^2-4x+5}=\int_{b}^{a}\frac{dx}{\left ( \sqrt{2}x-\sqrt{2} \right )^2+3}\\t=\frac{\sqrt{2}(x-1)}{\sqrt{3}}\Rightarrow dx=\frac{\sqrt{3}}{\sqrt{2}}dt\\ \frac{1}{\sqrt{6}}\int \frac{dt}{t^2+1}= \frac{1}{\sqrt{6}}arctg(t)+C\Rightarrow \left.\begin{matrix}\frac{1}{\sqrt{6}}arctg\left ( \frac{\sqrt{2}(x-1)}{\sqrt{3}} \right )\end{matrix}\right|^a_b\\\frac{1}{\sqrt{6}}arctg\left ( \frac{\sqrt{2}(a-1)}{\sqrt{3}} \right )-\frac{1}{\sqrt{6}}arctg\left ( \frac{\sqrt{2}(b-1)}{\sqrt{3}} \right)$