Question

15题，求定积分，要过程谢谢。

Answer 1

令x=sint，则dx=costdt
原式=∫(π/4,π/2) cost/sin^2t*costdt
=∫(π/4,π/2) cot^2tdt
=∫(π/4,π/2) (csc^2t-1)dt
=(-cott-t)|(π/4,π/2)
=-π/2+1+π/4
=1-π/4

Answer 2

设x=sinθ∈(1/√2,1)
θ∈(π/4,π/2)
J=∫{π/4,π/2} cosθ/(sinθ)^2 d(sinθ)
=∫{π/4,π/2}1-1/(sinθ)^2 dθ
=π/4+1/tan(π/2)-1/tan(π/4)
=π/4-1

追答

修正

设x=sinθ∈(1/√2,1)
θ∈(π/4,π/2)
J=∫{π/4,π/2} cosθ/(sinθ)^2 d(sinθ)
=∫{π/4,π/2}-1+1/(sinθ)^2 dθ
=-π/4-1/tan(π/2)+1/tan(π/4)
=1-π/4

追问

倒数第三行的负1 哪来的

追答

追问

非常感谢

追答

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