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Integrate the function$\frac{(\large log \; x)^2}{\large x}$

This question has appeared in model paper 2012

1 Answer

  • Method of substitution:
  • Given $\int f(x)dx$ can be transformed into another form by changing independent variable x to t by substituting x=g(t).
  • Consider $I=\int f(x)dx.$
  • Put x=g(t) so that $\frac{dx}{dx}=g'(t).
  • dx=g'(t)dt.
  • Thus $ I=\int f(g(t).g'(t))dt.$
Given $I=\int \frac{(log x)^2}{x}dx.$-------(1)
Let us substitute log x=t.
Differentiating on both sides we get,
Now substituting for log x and $\frac{1}{x}dx$ we get,
$I=\int t^2.dt$
On integrating we get,
Substituting back for t we get,
Hence $\int\frac{(log x)^2}{x}dx=\frac{1}{3}(log|x|)^3+c$.


answered Jan 28, 2013 by sreemathi.v
edited Jan 28, 2013 by sreemathi.v