Question

Integrate the given equation $\int \frac{\sqrt{x^2+1}(\log (x^2+1)-2\log x)}{x^4}dx$

• A. $\frac{-1}{3}(1+\frac{1}{x^2}{)}^{\frac{3}{2}}\log (1+\frac{1}{x^2})+\frac{2}{9}(1+\frac{1}{x^2}{)}^{\frac{3}{2}}+c$
• B. $\frac{-1}{3}{\left(1+\frac{1}{x^3}\right)}^{\frac{3}{2}}\log \left(1+\frac{1}{x^3}\right)+\frac{2}{9}{\left(1+\frac{1}{x^3}\right)}^{\frac{3}{2}}+c$
  
• C. $\frac{-1}{9}(1+\frac{1}{x^2}{)}^{\frac{3}{2}}\log (1+\frac{1}{x^2})+\frac{2}{9}(1+\frac{1}{x^2}{)}^{\frac{3}{2}}+c$
• D. None of these

Answer 1

The correct option is B $\frac{-1}{3}{\left(1+\frac{1}{x^3}\right)}^{\frac{3}{2}}\log \left(1+\frac{1}{x^3}\right)+\frac{2}{9}{\left(1+\frac{1}{x^3}\right)}^{\frac{3}{2}}+c$

${\int }^{}\frac{\sqrt{x^2+1}\left(\log \left(x^2+1\right)-2\log x\right)}{x^4}dx$

$={\int }^{}\frac{x\sqrt{1+\frac{1}{x^2}}\left(\log \frac{x^2+1}{x^2}\right)}{x\times x^3}dx$

${\int }^{}\sqrt{1+\frac{1}{x^2}}\log \left(1+d\frac{1}{x^2}\right)\times \frac{1}{x^3}dx$

Putting $1+\frac{1}{x^3}=t$

$\Rightarrow \frac{-2}{x^3}dx=dt$

$\Rightarrow \frac{dx}{x^3}=\frac{-dt}{2}$

$=\frac{-1}{2}{\int }^{}\sqrt{t}\log tdt$

$=\frac{-1}{2}\left[\log t\times \frac{2}{3}{t}^{\frac{3}{2}}-{\int }^{}\frac{1}{t}\times \frac{2}{3}{t}^{\frac{3}{2}}dt\right]$

$=\frac{-1}{2}\left[\frac{2}{3}{t}^{\frac{3}{2}}\log t-\frac{2}{3}{\int }^{}{t}^{\frac{1}{2}}dt\right]$

$=\frac{-1}{3}\left[t^{\frac{3}{2}}\log t-\frac{2}{3}{t}^{\frac{3}{2}}\right]+c$

$=\frac{-1}{3}{\left(1+\frac{1}{x^3}\right)}^{\frac{3}{2}}\log \left(1+\frac{1}{x^3}\right)+\frac{2}{9}{\left(1+\frac{1}{x^3}\right)}^{\frac{3}{2}}+c$