Question

${\int }_1^2\frac{d_X}{x+x^3}=$

• A. $\frac{1}{2}\log \frac{8}{5}$
• B. $\frac{1}{3}\log \frac{8}{5}$
• C. $\frac{1}{2}\log \frac{5}{8}$
• D. $\frac{1}{3}\log \frac{5}{8}$

Answer 1

Answer: (A)

$\frac{1}{2}\log \frac{8}{5}$

Given, ${\int }_1^2\frac{dx}{x+x^3}$
$\Rightarrow$ ${\int }_1^2\frac{dx}{x(1+x^2)}$
Breaking the above equation in partial fraction we have :

$\Rightarrow$ ${\int }_1^2[\frac{1}{x}-\frac{x}{1+x^2}]dx$
$\Rightarrow$ ${\int }_1^2[\frac{1}{x}-\frac{1}{2}\frac{2x}{1+x^2}]dx$
$\Rightarrow$ $[\frac{1}{2}\log 4-\frac{1}{2}\log 5]-[\frac{1}{2}\log 1-\frac{1}{2}\log 2]$
$\Rightarrow$ $\frac{1}{2}\log \frac{8}{5}$ is the answer.