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Home  >>  JEEMAIN and AIPMT  >>  Mathematics  >>  Class12  >>  Integral Calculus
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Integrate : $\int \sqrt {x^2+1} \log \large\frac{(x^2+1)-2 \log x }{x^4}$$dx$

$(a)\;\frac{-1}{2}\bigg \{ \frac{2}{3} (1+\frac{1}{x^2})-(1+\frac{1}{x^2})^{3/2} \times \frac{4}{9}\bigg\}+c \\(b)\;\frac{1}{2}\bigg \{ \frac{2}{3} (1+\frac{1}{x^2})-(1+\frac{1}{x^2})^{3/2} \times \frac{4}{9}\bigg\}+c \\(c)\;\frac{-1}{2}\bigg \{ \frac{5}{3} (1+\frac{1}{x^2})-(1+\frac{1}{x^2})^{3/2} \times \frac{4}{9}\bigg\}+c \\ (d)\;None$

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1 Answer

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$\int \sqrt{\large\frac{1+x}{x^2}} . \{\log (1+ \large\frac{1}{x^2}) \} \times \large\frac{1}{x^3} $$dx$
$1+\large\frac{1}{x^2}$$=t$
$0- \large\frac{2}{x^3}$$dx=dt$
=> $\int \sqrt{t} \log (t) \times \large\frac{-dt}{2}$
=> $-\large\frac{-1}{2}$$ \int \sqrt t . \log t .dt$
=> $ \large\frac{-1}{2}$$ \{ \log.t \times t^{3/2} \times \large\frac{2}{3} -\int \large\frac{t^{3/2}}{t} \times$$ \frac{2}{3}dt \}$
=> $ \large\frac{-1}{2}$$ \{ \large\frac{2}{3} $$ (t)^{3/2} \log (t) - (t)^{\frac{2}{3}} \times (\large\frac{2}{3})^2+c\}$
$\large\frac{-1}{2} \bigg \{\frac{2}{3} (1+\frac{1}{x^2})^{3/2} $$\log (1+\frac{1}{x^2}) -(1+\frac{1}{x^2})^{3/2} \times \frac{4}{9}\bigg\}+c $
Hence a is the correct answer.
answered Dec 30, 2013 by meena.p
 
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