Since D^{-1}(1+x^2)^{-1/2}=\sinh^{-1}(x)+C, is it ture that \sinh^{-1}x+C|_0^1=\log(1+\sqrt{2})?

broggesy9

broggesy9

Answered question

2022-04-25

Since
D1(1+x2)12=sinh1(x)+C,
is it ture that
sinh1x+C01=log(1+2)?

Answer & Explanation

Adrien Ho

Adrien Ho

Beginner2022-04-26Added 16 answers

A yet different approach:
Let x=tany to obtain
01dx1+x2=0π4secydy
=0π4secysecy+tanysecy+tanydy
=0π4sec2y+secytanysecy+tanydy
=0π4d(secy+tany)secy+tany
=log|secy+tany|y=0y=π4
=log|secπ4+tanπ4|log|sec0+tan0|
=log|2+1|log|1+0|=log(2+1)

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