If the value of the integral ∫(from 0 to 1/2) x²/(1-x²)³/² dx is k/6, then k is equal to:

2√3 - π

2√3 + π

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Integration by Substitution Maths Integrals Maths Class 12th

If the value of the integral ∫(from 0 to 1/2) x²/(1-x²)³/² dx is k/6, then k is equal to:

Option 1 -

2√3 + π

Option 2 -

3√2 - π

Option 3 -

2√3 - π

Option 4 -

3√2 + π

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1 Answer

Answered by

alok kumar singh | Contributor-Level 10

a month ago

Correct Option - 3

Detailed Solution:

k/6 = ∫? ^ (π/6) (x²)/ (1-x²)³/² dx x = sinθ dx = cosθdθ
=> k/6 = ∫? ^ (π/6) (sin²θ)/ (1-sin²θ)³/² * cosθdθ
=> k/6 = ∫? ^ (π/6) (sin²θ)/ (cos³θ) * cosθdθ
=> k/6 = ∫? ^ (π/6) tan²θdθ = ∫? ^ (π/6) (sec²θ-1)dθ
=> k/6 = [tanθ - θ]? ^ (π/6) = (1/√3 - π/6) = (2√3-π)/6
=> k = 2√3 - π

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