计算下列积分∫上限是2下限是1(根号下(x^2-1)dx/x)
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计算下列积分∫上限是2下限是1(根号下(x^2-1)dx/x)
令x = secz,dx = secztanz dz
x > 1,z ∈[0,π/2),√(x² - 1) = √(sec²z - 1) = tanz
∫(1~2) √(x² - 1)/x dx
= ∫(0~π/3) tanz/secz * secztanz dz
= ∫(0~π/3) tan²z dz
= ∫(0~π/3) (sec²z - 1) dz
= [tanz - z] |(0~π/3)
= [tan(π/3) - π/3] - [tan(0) - 0]
= √3 - π/3
x > 1,z ∈[0,π/2),√(x² - 1) = √(sec²z - 1) = tanz
∫(1~2) √(x² - 1)/x dx
= ∫(0~π/3) tanz/secz * secztanz dz
= ∫(0~π/3) tan²z dz
= ∫(0~π/3) (sec²z - 1) dz
= [tanz - z] |(0~π/3)
= [tan(π/3) - π/3] - [tan(0) - 0]
= √3 - π/3
计算下列积分∫上限是2下限是1(根号下(x^2-1)dx/x)
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