Mathematics

If $$\displaystyle\int \dfrac{(x-1)dx}{x^{2}\sqrt{2x^{2}-2x+1}}=\dfrac{\sqrt{f(x)}}{g(x)}+C$$, where $$f(x)$$ is a quadratic expression and $$g(x)$$ is a monic linear expression.


ANSWER

$$f(x)=2x^{2}-2x+1$$


SOLUTION
Let $$I=\displaystyle \int{\dfrac{(x-1)dx}{x^2\sqrt{2x^2-2x+1}}}=\displaystyle \int{\dfrac{1/x-1/x^2}{\sqrt{2x^2-2x+1}}}dx$$

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

put 
$$2-\dfrac{2}{x}+\dfrac{1}{x^2}=t$$

$$\left(\dfrac{2}{x^2}-\dfrac{2}{x^3}\right)dx=dt$$

$$I=\dfrac{1}{2}\displaystyle \int{\dfrac{dt}{\sqrt t}}=\sqrt t+c$$
put $$t=2-\dfrac{2}{x}+\dfrac{1}{x^2}$$

$$I=\sqrt{2-\dfrac{2}{x}+\dfrac{1}{x^2}}+c=\dfrac{\sqrt{2x^2-2x+1}}{x}+c$$

$$f(x)=2x^2-2x+1$$

$$g(x)=x$$
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Single Correct Medium Published on 17th 09, 2020
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