Question

Let $f(x)=\int \frac{2 x}{\left(x^2+1\right)\left(x^2+3\right)} d x$ If $f(3)=\frac{1}{2}\left(\log _e 5-\log _e 6\right)$, then $f(4)$ is equal to

• A. $\log _{ e } 17-\log _{ e } 18$
• B. $\log _e 19-\log _e 20$
• C. $\frac{1}{2}\left(\log _e 19-\log _e 17\right)$
• D. $\frac{1}{2}\left(\log _e 17-\log _e 19\right)$

Answer 1

The Correct Option is D

Put $x^2=t$
$?\frac{dt}{(t+1)(t+3)}=\frac{1}{2}?\left(\frac{1}{t+1}-\frac{1}{t+3}\right)dt$
$f(x)=\frac{1}{2}\ln \left(\frac{x^2+1}{x^2+3}\right)+C$
$f(3)=\frac{1}{2}(\ln 10-\ln 12)+C$
$?C=0$
$f(4)=\frac{1}{2}\ln \left(\frac{17}{19}\right)$