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Evaluate : $ \int\large\frac{(log\: x)^2}{x} $$dx$

1 Answer

  • Method of substitution :
  • Given 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 $\large\frac{dx}{dt}$$=g'(t).$
  • $\Rightarrow $dx=g'(t)dt.
  • Thus $I=\int f(g(t).g'(t))dt.$
Step 1:
$I=\int\large\frac{(\log x)^2}{x}$$dx$
Let $\log x=t$
On differentiating we get,
$I=\int t^2dt$
Step 2:
On integrating we get,
Substituting for $t$ we get,
$I=\large\frac{(\log x)^3}{3}$$+c$
answered Oct 2, 2013 by sreemathi.v