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Question Number 122430 by benjo_mathlover last updated on 17/Nov/20

∫ (x^3 /( (√(4−x^2 ))+x^2 −4)) dx

∫x34−x2+x2−4dx

Answered by liberty last updated on 17/Nov/20

let (√(4−x^2 )) = ω ∧ x^2 −4 =−ω^2 ⇒x dx =−ω dω Q(x)=∫(((4−ω^2 )(−ω dω))/(ω−ω^2 )) Q(x)= ∫ (((ω^2 −4)/(1−ω))) dω Q(x)= ∫ (−ω−1−(3/(1−ω))) dω Q(x)=−(1/2)ω^2 −ω+3ln ∣ω−1∣+c Q(x)=−(1/2)(4−x^2 )−(√(4−x^2 )) + 3ln ∣(√(4−x^2 ))−1∣ + c

let4−x2=ω∧x2−4=−ω2⇒xdx=−ωdωQ(x)=∫(4−ω2)(−ωdω)ω−ω2Q(x)=∫(ω2−41−ω)dωQ(x)=∫(−ω−1−31−ω)dωQ(x)=−12ω2−ω+3ln∣ω−1∣+cQ(x)=−12(4−x2)−4−x2+3ln∣4−x2−1∣+c

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