Integration sUBSTITion (1 Viewer)

[SBHS4EVER]

THis is in Patel's blue book-

the integral of 1/ sqrt x times srqt(1 - x)

HOW??????

 

[wogboy]

a = integral sqrt(1 - x)/sqrt(x) dx

so let x = cos^2(@)

dx/d@ = -2sin@cos@

dx = -2sin@cos@ d@

so,

a = integral sqrt(1 - x)/sqrt(x) * (-2sin@cos@) d@
= integral sqrt(1 - cos^2(@))/sqrt(cos^2(@)) * (-2sin@cos@) d@
= integral sqrt(tan^2(@)) * (-2sin@cos@) d@
= integral -2sin@cos@tan@ d@
= integral -2sin^2(@) d@

now use the identity, sin^2(@) = 1/2 - (cos2@)/2

so

a = integral -2(1/2 - (cos2@)/2) d@
= integral (-1 + cos2@) d@
= -@ + (sin2@)/2 + C
= -@ + sin@cos@ + C
= -@ + cos@sqrt(1 - cos^(@)) + C

now @ = acos(sqrt(x))

{acos means inverse cosine}

so,

a = -acos(sqrt(x)) + sqrt(x) * sqrt (1 - x) + C
= -acos(sqrt(x)) + sqrt(x/(1-x)) + C