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

Answer: (D)

$\frac{1}{2}\left(\log _e 17-\log _e 19\right)$

Answer 2

Put x2=t
?(t+1)(t+3)dt?=21??(t+11?-t+31?)dt
f(x)=21?ln(x2+3x2+1?)+C
f(3)=21?(ln10-ln12)+C
?C=0
f(4)=21?ln(1917?)