Integrate the function in Exercise: 1 (x-a)(x-b) a - Vidyadip

Question

Integrate the function in Exercise:
$\frac{1}{\sqrt{(\text{x}-\text{a})(\text{x}-\text{b})}}$a

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Answer

$\int\frac{1}{\sqrt{(\text{x}-\text{a})(\text{x}-\text{b})}}\text{ dx}$
$=\int\frac{1}{\sqrt{\text{x}^2-\text{bx}-\text{ax}+\text{ab}}}\text{ dx}$
$=\int\frac{1}{\sqrt{\text{x}^2-\text{x}(\text{a}+\text{b})+\text{ab}}}\text{ dx}$
$=\int\frac{1}{\sqrt{\text{x}^2-\text{x}(\text{a}+\text{b})+\bigg(\frac{\text{a}+\text{b}}{2}\bigg)^2-\bigg(\frac{\text{a}+\text{b}}{2}\bigg)^2+\text{ab}}}\text{ dx}$
$=\int\frac{1}{\sqrt{\Bigg[\bigg\{\text{x}-\Big(\frac{\text{a+b}}{2}\Big)\bigg\}^2-\bigg\{\frac{(\text{a+b})^2}{4}-\text{ab}\bigg\}\Bigg]}}\text{ dx}$
$=\int\frac{1}{\sqrt{\Bigg[\bigg\{\text{x}-\Big(\frac{\text{a+b}}{2}\Big)\bigg\}^2-\bigg\{\frac{(\text{a+b})^2-4\text{ab}}{4}\bigg\}\Bigg]}}\text{ dx}$
$=\int\frac{1}{\sqrt{\Bigg[\bigg\{\text{x}-\Big(\frac{\text{a+b}}{2}\Big)\bigg\}^2-\bigg\{\frac{(\text{a}-\text{b})^2}{4}\bigg\}\Bigg]}}\text{ dx}$
$=\int\frac{1}{\sqrt{\Bigg[\bigg\{\text{x}-\Big(\frac{\text{a+b}}{2}\Big)\bigg\}^2-\bigg\{\bigg(\frac{\text{a}-\text{b}}{2}\bigg)^2\bigg\}\Bigg]}}\text{ dx}$
$=\log\begin{vmatrix}\text{x}-\bigg(\frac{\text{a}+\text{b}}{2}\bigg)+\sqrt{\bigg\{\Big(\text{x}-\frac{\text{a+b}}{2}\Big)^2-\bigg(\frac{\text{a}-\text{b}}{2}\bigg)^2\bigg\}}\end{vmatrix}+\text{c}$
$=\log\begin{vmatrix}\text{x}-\bigg(\frac{\text{a}+\text{b}}{2}\bigg)+\sqrt{\text{x}^2-\text{x}(\text{a}+\text{b})+\text{ab}}\end{vmatrix}+\text{c}$

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