Mathematics

Solve:
$$\int {\dfrac{{{x^3} - 4{x^2} + 6x + 5}}{{{x^2} - 2x + 3}}} dx$$


SOLUTION
$$\displaystyle \int \dfrac{x^3-4x^3+6x+5}{x^2-2x+3}dx$$

This can aslo be written as 

$$=\displaystyle \int x-2+\dfrac{11-x}{x^2-2x+3}dx$$

$$=\dfrac{x^2}{2}-2x+\displaystyle \int \dfrac{10}{x^2-2x+3}+\dfrac{-1}{2}\displaystyle \int \dfrac{2x-2}{x^2-2x+3}dx$$

$$=\dfrac{x^2}{2}-2x+\dfrac{10}{\sqrt{2}}\tan^{-1}\left(\dfrac{x-1}{\sqrt{2}}\right)-\dfrac{1}{2}\log|x^2-2x+3|+c$$

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Subjective Medium Published on 17th 09, 2020
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