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Plz solve this integration.............. ∫ ((√((x)^2+x+1))/(x+1)) dx Sir in this, x 2 +x+1 = (x+1) 2 - x where x+1 is replaced by t then we get ((v((t)^2-(t-1)))/t) dt right; then how to proceed sir..... where 'v' represents sq.rt


Plz solve this integration..............


     ∫ ((√((x)^2+x+1))/(x+1)) dx



 


 


Sir in this,


 x2+x+1 = (x+1)2 - x


where x+1 is replaced by t


then we get


((v((t)^2-(t-1)))/t) dt right; then how to proceed sir.....







where 'v' represents sq.rt

Grade:11

1 Answers

Jitender Singh IIT Delhi
askIITians Faculty 158 Points
8 years ago
Ans:
I = \int \frac{\sqrt{x^{2}+x+1}}{x+1}dx
I = \int \frac{\sqrt{(x+\frac{1}{2})^{2}+\frac{3}{4}}}{x+1}dx
u = x + \frac{1}{2}
du = dx
I = \int \frac{\sqrt{(u)^{2}+\frac{3}{4}}}{u+\frac{1}{2}}du
u = \frac{\sqrt{3}}{2}tan(s)
du = \frac{\sqrt{3}}{2}sec^{2}(s).ds
I = \frac{3}{4}\int \frac{sec^{3}(s)}{\frac{\sqrt{3}}{2}tan(s)+\frac{1}{2}}ds
t = tan (\frac{s}{2})
dt = \frac{1}{2}sec^{2} (\frac{s}{2}).ds
I = 3\int \frac{(p^{2}+1)^{2}}{(p^{2}-1)^{2}.(-p^{2}+2\sqrt{3}p+1)}dp
Use simply partial fraction rule here, you will get
I = \sqrt{x^{2}+x+1}-log(2 \sqrt{x^{2}+x+1}-(x-1))+log(x+1)-\frac{1}{2}sinh^{-1}(\frac{2x+1}{\sqrt{3}}) + c
Thanks & Regards
Jitender Singh
IIT Delhi
askIITians Faculty

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