Question
Transcribed Text
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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