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Hi ,
∫√((5-x)/(x-2)) dx
take , √(x-2) = y
x = y2 +2
dx = 2y dy
2∫√(5- y2 -2)/ y *y dy =2∫√(3- y2) dy
= 2√3 ∫(1- (y/√3)2 ) dy
y/√3 = sin z
dy = √3 cos z dz
= 2√3 ∫(1- sin2 z ) cos z dz
= 2√3 ∫ cos 2z dz
= √3 ∫( 1+cos2z) dz
= √3 [z + sin 2z/2 ]
= √3 [sin ⁻¹( (√ (x-2)/3) + cos (sin ⁻¹( √( (x-2)/3)) +√ ((x-2)/3) +K
Hope it helps you !
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