InterviewSolution
Saved Bookmarks
InterviewSolution
| 1. |
\( \int_{\pi}^{0} \frac{\sqrt{x^{2}+1}\left(\log \left(x^{2}+1\right)-2 \log x\right)}{x^{4}} d x \) |
|
Answer» Let I = \(\int\cfrac{\sqrt{x^2+1}(log(x^2+1)-2 log x)}{x^4}dx\) \(=\int\cfrac{x\sqrt{1+\frac1{x^2}}log(\frac{x^2+1}{x^2})}{x^4}dx\) (\(\because\) 2log x = log x2 and log(x2 + 1) - 2logx = log(x2 + 1) - logx2 = log(\(\frac{x^2+1}{x^2}\))) \(=\int\cfrac{\sqrt{1+\frac1{x^2}}log(1+\frac1{x^2})}{x^3}dx\) Let 1 + \(\frac1{x^2}\) = t ⇒ \(\frac{-2}{x^3}dx = dt\) ⇒ \(\frac{dx}{x^3} = -\frac{dt}2\) \(\therefore\) I = \(-\frac12\int\sqrt t\) log t dt \(=-\frac12[log t\int \sqrt t dt-\int(\frac{d}{dt}log t\int\sqrt t dt)dt]\) \(=-\frac12[\frac23log t t^{3/2}- \int\frac1t\times\frac23t^{3/2}dt]\) \(=-\frac12[\frac23t^{3/2}log t - \frac23\int t^{1/2}dt]\) \(= -\frac12(\frac23t^{3/2}log t - \frac23\times\frac23t^{3/2})\) \(=-\frac12[\frac23(1+\frac1{x^2})^{3/2}log(1+\frac1{x^2}-\frac49(1+\frac1{x^2})^{3/2}]\) (By putting t = 1 + \(\frac1{x^2}\)) |
|