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what is the integration of the following statement ∫sqrt(1+ 1/x) dx

what is the integration of the following statement
∫sqrt(1+ 1/x) dx

Grade:12th pass

1 Answers

Aditya Gupta
2075 Points
2 years ago
let u=sqrt(1+ 1/x) 
or u^2= (1+ 1/x)
2udu= – dx/x^2
so integral becomes udx= –u*2ux^2du= – 2(ux)^2du= – 2u^2(1/(u^2 – 1))^2du
now u^2(1/(u^2 – 1))^2= u^2/(u^2 – 1)^2= u^2/(u – 1)^2(u+1)^2, which can be easily solved using partial fractions.
u^2/(u – 1)^2(u+1)^2= A/(u-1)+B/(u-1)^2+C/(u+1)+D/(u+1)^2, now determine the coeffs A, B, C, D in the usual way and you are good to go!

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