Mathematics

Evaluate the following integral:
$$\displaystyle \int { \cfrac { 1 }{ \sqrt { 5{ x }^{ 2 }-2x }  }  } dx$$


SOLUTION
$$\displaystyle\int{\dfrac{dx}{\sqrt{5{x}^{2}-2x}}}$$

We have $$5{x}^{2}-2x=5\left({x}^{2}-2\times\dfrac{1}{5}x+\dfrac{1}{25}-\dfrac{1}{25}\right)$$

$$=\displaystyle\int{\dfrac{dx}{\sqrt{5\left({x}^{2}-2\times\dfrac{1}{5}x+\dfrac{1}{25}-\dfrac{1}{25}\right)}}}$$

$$=\dfrac{1}{\sqrt{5}}\displaystyle\int{\dfrac{dx}{\sqrt{\left({x}^{2}-2\times\dfrac{1}{5}x+\dfrac{1}{25}-\dfrac{1}{25}\right)}}}$$

$$=\dfrac{1}{\sqrt{5}}\displaystyle\int{\dfrac{dx}{\sqrt{\left({\left(x-\dfrac{1}{5}\right)}^{2}-{\left(\dfrac{1}{5}\right)}^{2}\right)}}}$$

We know that $$\displaystyle\int{\dfrac{dx}{\sqrt{{x}^{2}-{a}^{2}}}}=\log{\left|x+\sqrt{{x}^{2}-{a}^{2}}\right|}+c$$

Replace $$x\rightarrow\,x-\dfrac{1}{5}$$ and $$a\rightarrow\,\dfrac{1}{5}$$ we get

$$=\dfrac{1}{\sqrt{5}}\log{\left|x-\dfrac{1}{5}+\sqrt{{\left(x-\dfrac{1}{5}\right)}^{2}-\dfrac{1}{25}}\right|}+c$$        

$$=\dfrac{1}{\sqrt{5}}\log{\left|\dfrac{5x-1}{5}+\sqrt{{\left(\dfrac{5x-1}{5}\right)}^{2}-\dfrac{1}{25}}\right|}+c$$ 

$$=\dfrac{1}{\sqrt{5}}\log{\left|\dfrac{5x-1}{5}+\sqrt{\dfrac{{\left(5x-1\right)}^{2}}{25}-\dfrac{1}{25}}\right|}+c$$  

$$=\dfrac{1}{\sqrt{5}}\log{\left|\dfrac{5x-1}{5}+\sqrt{\dfrac{25{x}^{2}-10x+1-1}{25}}\right|}+c$$    

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