Mathematics

$$\displaystyle\int \dfrac{2x}{2+x-x^2}dx$$.


SOLUTION
$$\displaystyle=\int\dfrac{2x-1+1}{2+x-x^2}dx$$                             

$$\displaystyle\int\dfrac{2x-1}{2+x-x^2}dx+\int\dfrac{dx}{2+x-x^2}$$          

$$\displaystyle=\int\dfrac{-dz}{z}-\int\dfrac{dx}{x^2-x-2}$$                   

$$\displaystyle2-\log|z|-\int\dfrac{dx}{x^2-2;\dfrac{x}{2}+(\dfrac{1}{2})^2-2-(\dfrac{1}{2})^2}$$

$$\displaystyle=-\log|z+x-x^2|-\int\dfrac{dx}{{x-\dfrac{1}{2}}^2-{\dfrac{\sqrt{5}}{2}}^2}$$

$$=-\log|z+x-x^2|-\dfrac{\sqrt{2}}{2.\sqrt{5}}\log \left|\dfrac{x-\dfrac{1}{2}-\dfrac{\sqrt{5}}{2}}{x-\dfrac{1}{2}+\dfrac{\sqrt{5}}{2}}\right|+c$$

$$=-\log |z+x-x^2|-\dfrac{1}{\sqrt{10}}\log\left|\dfrac{x-1-\sqrt{5}}{x-1+\sqrt{5}}\right|+c$$
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Subjective Medium Published on 17th 09, 2020
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