Question

求∫（1→根号3）1／x＾2根号1＋x＾2

Answer 1

问：求∫（1→根号3）1／x＾2根号1＋x＾2

答：此题用三角代换（换元法）令x=tant, 则 dx=sec²tdt∵x∈[1,√3]∴不妨令t∈[π/4,π/3](在此区间上,x随t单增,sect≥0)原积分=∫(π/4,π/3) sec²tdt/(tan²t·sect)=∫(π/4,π/3) sectdt/tan²t=∫(π/4,π/3) dt/(tan²t·cost)=∫(π/4,π/3) costdt/sin²t=∫(π/4,π/3) d(sint)/sin²t=[-1/sint]|(π/4,π/3)=√2-2/√3=√2-√6/3希望我的解答对你有所帮助