I would like to ask for some help regarding the following indefinite integral, tried integration by parts and trigonometric substitution which both brought me to ∫secθtanθdθ, and from this point it is messy to integrate by parts, any help would be appreciated.
∫dxx√x2+1
Answer
∫secθtanθ=1cosθsinθcosθdθ=∫1sinθdθ=∫cscθdθ
Alternatively, given ∫dxx√x2+1=∫xdxx2√x2+1
Put x2+1=u2⟺x2=u2−1⟹udu=xdx
This gives us the integral, after substitution: ∫udu(u2−1)u=∫du(u2−1)=12∫(1u−1−1u+1)du
I'm sure you can take it from here.
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