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
Questions 203525
Subjects 9
Chapters 126
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