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Integrate the function\[\frac{(x+1)(x+log\: x)^2}{x}\]

$\begin{array}{1 1} \frac{(x+log x)^3}{3}+c \\ \frac{(1+log x)^3}{3}+c \\ \frac{(x+log x)^2}{3}+c \\ \frac{(x+log x)^2}{3}+c \end{array}$

1 Answer

  • (i)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 $\frac{dx}{dt}=g'(t).$
  • $\Rightarrow $dx=g'(t)dt.
  • Thus $I=\int f(g(t).g'(t))dt.$
  • (ii)$\int \frac{1}{x}dx=log x+c.$
Given $I=\int \frac{(x+1)(x+log x)^2}{x}dx=\int (1+\frac{1}{x})(x+log x)^2dx$.
Put (x+log x)=t
Now substituting of x and dx we get,
$I=\int t^2.dt.$
On integrating we get,
Substituting back for t we get,
$\int\frac{(x+1)(x+log x)^2}{x}dx=\frac{(x+log x)^3}{3}+c.$


answered Jan 29, 2013 by sreemathi.v