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Home  >>  CBSE XII  >>  Math  >>  Model Papers
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Evaluate :$\int \large \frac{dx}{x(x^3+1)}$

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$\Large \int \frac{dx}{x(x^3+1)}$
Multiply and divide by $x^2$
$\Large \int \frac{x^2dx}{x^3(x^3+1)}$-----(1)
Put $x^3+1=t\Rightarrow x^3=t-1$
$3x^2dx=dt.$
$x^2dx=\Large \frac{dt}{3}$
Substitute this in equ(1)
$\Large \int\frac{dt/3}{(t-1)t}=\frac{1}{3}\int\frac{dt}{(t-1)(t)}$
$\Large \frac{1}{t(t-1)}=\frac{A}{t}+\frac{B}{t-1}$
1=A(t-1)+Bt
Put t=1.
B=1
Put t=0.
A=-1
$\Large \frac{1}{3}\int\frac{dt}{(t-1)(t)}=\frac{1}{3}\begin{bmatrix}\int\frac{-1}{t}+\frac{1}{(t-1)}\end{bmatrix}dt$
$\qquad\qquad=\Large \frac{-1}{3}\frac{dt}{t}+\frac{1}{3}\int\frac{dt}{(t-1)}$
$\qquad\qquad=\Large \frac{1}{3}log\frac{|t-1|}{t|}$+c.
Substituting for t
$\qquad\qquad=\Large \frac{1}{3}log\frac{|x^3+1-1|}{x^3+1|}$+c.
$\qquad\qquad=\Large \frac{1}{3}log\frac{|x^3|}{x^3+1|}$+c.
answered Mar 25, 2013 by sreemathi.v
 
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