# Integrate : $\int \sqrt {x^2+1} \log \large\frac{(x^2+1)-2 \log x }{x^4}$$dx (a)\;\frac{-1}{2}\bigg \{ \frac{2}{3} (1+\frac{1}{x^2})-(1+\frac{1}{x^2})^{3/2} \times \frac{4}{9}\bigg\}+c \\(b)\;\frac{1}{2}\bigg \{ \frac{2}{3} (1+\frac{1}{x^2})-(1+\frac{1}{x^2})^{3/2} \times \frac{4}{9}\bigg\}+c \\(c)\;\frac{-1}{2}\bigg \{ \frac{5}{3} (1+\frac{1}{x^2})-(1+\frac{1}{x^2})^{3/2} \times \frac{4}{9}\bigg\}+c \\ (d)\;None ## 1 Answer Need homework help? Click here. \int \sqrt{\large\frac{1+x}{x^2}} . \{\log (1+ \large\frac{1}{x^2}) \} \times \large\frac{1}{x^3}$$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