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Pattern - 37 of Indefinite Integration

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⋆ Pattern – 37

$\int\frac{1}{\sqrt{\left(x-a\right)\left(b-x\right)}}dx,\ \int\sqrt{\frac{x-a}{b-x}}\ dx$ , then put x = a $cos^2{\theta\ } + b \sin^2{\theta\ }$

1) Evaluate : $\int\sqrt{\frac{x-3}{4-x}}\ dx$

solution : Let, I = $\int\sqrt{\frac{x-3}{4-x}} dx$

Here, a = 3, b = 4

Let, x = 3 $\cos^2{\theta\ }+4\sin^2{\theta\ }$

= $\left(3\cos^2{\theta\ }+3\sin^2{\theta\ }\right)+\sin^2{\theta\ }$

=3+ $\sin^2{\theta\ }$

∴ dx = 2 sin θ ∙ cos θ dθ

$\therefore\ \ I=\ \int\sqrt{\frac{\sin^2{\theta\ }}{4-3-\sin^2{\theta\ }}}\times2\sin{\theta\ }\cos{\theta\ }\ d\theta\$

$=\ \int\frac{\sin{\theta\ }}{\sqrt{1-\sin^2{\theta\ }}}\times2\sin{\theta\ }\cos{\theta\ }\ d\theta\$

$=\ \int\frac{2\sin^2{\theta\ }}{\cos{\theta\ }}\times\cos{\theta\ }\ d\theta\$

= ∫ ( 1 – cos 2θ ) dθ

$=\ \theta-\ \frac{\sin{2\theta\ }}{2}+c$

x = 3 + $\sin^2{\theta\ }$ —–(1)

& I = θ – $\frac{\sin{2\theta\ }}{2}$ + c —–(2)

from (1), x – 3 =

∴ sin θ = $\sqrt{x-3}$

cos θ = $\sqrt{4-x}$

$\frac{\sin{2\theta\ }}{2}$

$=\ \frac{2\sin{\theta\ }\times \cos{\theta\ }}{2}$

$=\sqrt{\left(x-3\right)\left(4-x\right)}$

∴ from (2)

$I=\sin^{-1}{\sqrt{x-3}}-\ \sqrt{\left(x-3\right)\left(4-x\right)}+c$