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Home  >>  CBSE XII  >>  Math  >>  Integrals
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Integrate the function\[\frac{(x+1)(x+log\: x)^2}{x}\]

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

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1 Answer

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Toolbox:
  • (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
 
$(1+\frac{1}{x})dx=dt.$
 
Now substituting of x and dx we get,
 
$I=\int t^2.dt.$
 
On integrating we get,
 
$\frac{t^3}{3}+c$.
 
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
 
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