Tuesday, 8 July 2014

calculus - Integrate intfracdxxsqrtx2+1



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.



dxxx2+1


Answer



secθtanθ=1cosθsinθcosθdθ=1sinθdθ=cscθdθ



Alternatively, given dxxx2+1=xdxx2x2+1




Put x2+1=u2x2=u21udu=xdx
This gives us the integral, after substitution: udu(u21)u=du(u21)=12(1u11u+1)du



I'm sure you can take it from here.


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