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Question Number 115362 by Bird last updated on 25/Sep/20

calculate ∫_0 ^1 ((sinx)/(1+x^2 ))dx

calculate∫01sinx1+x2dx

Answered by TANMAY PANACEA last updated on 25/Sep/20

1≥sinx≥−1 (1/(1+x^2 ))≥((sinx)/(1+x^2 ))≥((−1)/(1+x^2 )) ∫_0 ^1 (dx/(1+x^2 ))≥∫_0 ^1 ((sinx)/(1+x^2 ))dx≥∫_0 ^1 ((−dx)/(1+x^2 )) tan^(−1) x∣_0 ^1 ≥I≥−tan^(−1) x∣_0 ^1 (π/4)≥I≥−(π/4)

1⩾sinx⩾−111+x2⩾sinx1+x2⩾−11+x2∫01dx1+x2⩾∫01sinx1+x2dx⩾∫01−dx1+x2tan−1x∣01⩾I⩾−tan−1x∣01π4⩾I⩾−π4

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