Question

Integrate the function.

No Copy / paste. Answer in the easy method. ​

Answer 1

$\underline{\textbf{Step-by-step explanation:}}$

let us consider,

$I=\sf\int {\frac{1}{\sqrt{(x-a) (x-b) }}} dx$

Now,$\sf (x - a)(x - b)=x^2- ax - bx + ab$

$\sf[x^2-(a+b)x+ab]$

$\sf[x^2 - (a+b)x ]=-ab$

add $\sf{(\frac{a+b}{2})}^{2}$ on both the sides

$\sf[x^2- (a+b)x +{(\frac{a+b}{2})}^{2}] = - ab + {(\frac{a+b}{2})}^{2}$

$\sf=[x^2 - (a + b)x +{(\frac{a+b}{2})}^{2}] + ab - {(\frac{a+b}{2})}^{2}$

$\sf={(x - \frac{a+b}{2})}^{2} + (\frac{4ab - a^2 - b^2 -2ab}{4})$

$\sf=(x-{\frac{a+b}{2}})^{2}-({\frac{a-b}{2}})^{2}$

therefore,

$\sf I=\int {\frac{dx}{\sqrt{({{x-\frac{a+b}{2}})^{2}-({\frac{a-b}{2}})^2}}}}$

$\sf=log |(x-\frac{a+b}{2})+\sqrt{({x-\frac{a+b}{2}})^{2}-({\frac{a-b}{2}})^2}|+c$

$\sf=log|(x -\frac{a+b}{2})+\sqrt{(x-a) (x-b) }|+c$