证明∫[π/10,x]sint²dt+∫[π/2,x]1/sint²dt=0在(π/10,π/2)内有唯一实根
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证明∫[π/10,x]sint²dt+∫[π/2,x]1/sint²dt=0在(π/10,π/2)内有唯一实根
证明∫[π/10,x]sint²dt+∫[π/2,x]1/sint²dt=0在(π/10,π/2)内有唯一实根
证明∫[π/10,x]sint²dt+∫[π/2,x]1/sint²dt=0在(π/10,π/2)内有唯一实根
令f(x)=∫[π/10,x]sint²dt+∫[π/2,x]1/sint²dt
在(π/10,π/2)
f'(x)=sin^2x+1/sin^2x>0
说明函数在(π/10,π/2)单增
f(π/10)=∫[π/2,π/10]1/sint²dt0
由介值定理得
f(x)在(π/10,π/2)内有唯一实根
即
∫[π/10,x]sint²dt+∫[π/2,x]1/sint²dt=0在(π/10,π/2)内有唯一实根