A friend who is preparing for her exam asked me such question. I thought it for quite a long time and finally I did it !

P.S. I think the point is using the sec !

t = √(u ^ 2  + 1)

dx = √(u ^ 2  + 1)/  u dt

∫ 1 /√(x ^ 2 + 1) dx

= ∫ (1 / t) * (√(u ^ 2  + 1) /  u) dt

= ∫   1 /  √(u ^ 2  + 1) dt

t = secθ

dt / dθ = tanθ * secθ

dt =  tanθ * secθ dθ

= ∫ 1 / √ ((secθ)^2 -1) * tanθ * secθ dθ

= ∫ 1 / tanθ  * tanθ * secθ dθ

=∫secθ dθ

=ln(secθ + tanθ) + C

=ln(t +  √(t ^ 2 – 1)) + C

=ln(√(u ^ 2 + 1) + u) + C

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