Question

The integral $16\underset{1}{\overset{2}{\int }}\frac{dx}{x^3{\left(x^2+2\right)}^2}$ is equal to

• A. $\frac{11}{6}-{\log }_{e}4$
• B. $\frac{11}{12}+{\log }_{e}4$
• C. $\frac{11}{6}+{\log }_{e}4$
• D. $\frac{11}{12}-{\log }_{e}4$

Answer 1

Answer: (C)

$I=16\underset{1}{\overset{2}{\int }}\frac{dx}{x^3{\left(x^2+2\right)}^2}$
$=16\underset{1}{\overset{2}{\int }}\frac{dx}{x^3{x}^4{\left(1+\frac{2}{x^2}\right)}^2}$
Let, $1+\frac{2}{x^2}=t\Rightarrow \frac{-4}{x^3}dx=dt$
$I=-4\underset{3}{\overset{\frac{3}{2}}{\int }}\frac{dt}{{\left(\frac{2}{t-1}\right)}^2{t}^2}$
$I=-4\underset{3}{\overset{\frac{3}{2}}{\int }}{\left(\frac{t-1}{2}\right)}^2\frac{dt}{t^2}$
$I=-\frac{4}{4}\underset{3}{\overset{\frac{3}{2}}{\int }}\left(1-\frac{2}{t}+\frac{1}{t^2}\right)dt$
$I=-1{\left[t-2\ell n|t|-\frac{1}{t}\right]}_3^{\frac{3}{2}}$
$I=-1\left[\left(\frac{3}{2}-2\ell n\frac{3}{2}-\frac{2}{3}\right)-\left(3-2\ell n3-\frac{1}{3}\right)\right]$
$I=-1\left[2\ell n2-\frac{11}{6}\right]$
$I=\frac{11}{6}-\ell n4$