91 If 𝑎 and 𝑏 are positive numbers, show that ∫ 𝑥 ...

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91 If 𝑎 and 𝑏 are positive numbers, show that ∫ 𝑥 𝑎(1 − 𝑥) 𝑏 𝑑𝑥 1 0 = ∫ 𝑥 𝑏(1 − 𝑥) 𝑎 𝑑𝑥 1 0 92 If 𝑓 is continuous on [0, 𝜋], use the substitution 𝑢 = 𝜋 − 𝑥 to show that ∫ 𝑥𝑓(sin 𝑥)𝑑𝑥 𝜋 0 = 𝜋 2 ∫ 𝑓(sin 𝑥) 𝜋 0 𝑑𝑥 93 Use Exercise 92 to evaluate the integral ∫ 𝑥 sin(𝑥) 1 + cos2(𝑥) 𝑑𝑥 𝜋 0 94 a If 𝑓 is continuous, prove that ∫ 𝑓(cos 𝑥)𝑑𝑥 𝜋⁄2 0 = ∫ 𝑓(sin 𝑥)𝑑𝑥 𝜋⁄2 0 b Use part (a) to evaluate ∫ 𝑓(cos2 𝑥)𝑑𝑥 𝜋⁄2 0 and ∫ 𝑓(sin2 𝑥)𝑑𝑥 𝜋⁄2 0

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