1.How will u integrate x^2/sqrt(1-x^2) 2. Integrate x/sqrt(1-x^2)

Asked by GeoffreyRichards | 19th Oct, 2010, 04:54: PM

Expert Answer:

Dear Student,
To evaluate, ∫x2 dx/√(1 - x2)
Put x = sinθ
dx = cosθ dθ
∫x2 dx/√(1 - x2) = ∫sin2θ cosθdθ/√(1 - sin2θ)
= ∫sin2θ cosθdθ/cosθ
= ∫sin2θ dθ 
= ∫(1 - cos2θ)dθ / 2                      .. using cos2θ = 1 - 2sin2θ
= (θ - sin2θ /2)/2 + c  
= (θ/2 - sin2θ /4 + c  
= (sin-1x)/2 - (sin2(sin-1x))/4 + c                          ... Answer Q1
Similarly the second becomes after same substitution,
  ∫x dx/√(1 - x2) =  ∫sinθ dθ 
= - cosθ + c
= - cos sin-1 x + c               .. Answer Q 2
regards,
Team,
TopperLearning.

Answered by  | 19th Oct, 2010, 09:59: PM

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