Mathematics

# Evaluate: $\displaystyle\int \dfrac{1}{\sqrt{(x-1)} \sqrt{(x-2)}}dx$

$\ln(|2\sqrt{x^2-3x+2}+2x-3|)+c$

##### SOLUTION
$\displaystyle\int \dfrac{1}{\sqrt{x-1}\times \sqrt{x-2}}dx$
$I=\displaystyle\int \dfrac{1}{\sqrt{x-1}\times \sqrt{x-2}}dx$
$=\displaystyle\int \dfrac{1}{\sqrt{x^2-2x-x+2}}dx$
$=\displaystyle\int \dfrac{1}{\sqrt{x^2-3x+2}}dx$
$I=\displaystyle\int \dfrac{1}{\sqrt{(x-3/2)^2-(\sqrt{5}/\sqrt{2})^2}}dx$
as we know
$\displaystyle\int \dfrac{dx}{\sqrt{x^2-a^2}}=log |x+\sqrt{x^2-a^2}|+c$
So, $I=log\left(\left|-\dfrac{3}{2}+\sqrt{(x-\dfrac{3}{2})^2-(\sqrt{5}/2)^2}\right|\right)$
$=log\left(\left|2x-3+2\sqrt{x^2-3x+2}\right|\right)+C_1$
$I=log\left(\left|2\sqrt{x^2-3x+2}+2x-3\right)\right|+C_1$
where $C_1=C-log 2$.

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Single Correct Medium Published on 17th 09, 2020
Questions 203525
Subjects 9
Chapters 126
Enrolled Students 86

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