Mathematics

$$\displaystyle\:\int \frac{1}{2x^{2}+x-1}\; dx$$ is


ANSWER

$$\displaystyle \frac{1}{3}\log \left | \frac{2x-1}{2\left ( x+1 \right )} \right |+C$$


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

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

$$\displaystyle= \dfrac{1}{2}\int \dfrac{1}{\left ( x+\dfrac{1}{4}\right )^{2}-\left ( \dfrac{3}{4} \right )^{2}} dx$$

$$\displaystyle= \dfrac{1}{2}.\dfrac{1}{2\left ( \dfrac{3}{4} \right )}\log \left | \dfrac{x+\dfrac{1}{4}-\dfrac{3}{4}}{x+\dfrac{1}{4}+\dfrac{3}{4}} \right |+C$$

$$\displaystyle = \dfrac{1}{3}\log \left | \dfrac{x-\dfrac{1}{2}}{x+1} \right |+C= \dfrac{1}{3}\log \left | \dfrac{2x-1}{2\left ( x+1 \right )} \right |+C$$
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Single Correct Medium Published on 17th 09, 2020
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