Indefinite Integration Question 1
Question 1 - 25 January - Shift 1
Let $f(x)=\int \frac{2 x}{(x^{2}+1)(x^{2}+3)} d x$.If $f(3)=\frac{1}{2}(\log _{e} 5-\log _{e} 6)$, then $f(4)$ is equal to
(1) $\frac{1}{2}(\log _{e} 17-\log _{e} 19)$
(2) $\log _{e} 17-\log _{e} 18$
(3) $\frac{1}{2}(\log _{e} 19-\log _{e} 17)$
(4) $\log _{e} 19-\log _{e} 20$
Show Answer
Answer: (1)
Solution:
Formula: Integration by Substitution, Standard formula: Integration of $\int \frac{d x}{a x+b}$,
Put $x^{2}=t$
$\int \frac{d t}{(t+1)(t+3)}=\frac{1}{2} \int(\frac{1}{t+1}-\frac{1}{t+3}) d t$
$f(x)=\frac{1}{2} \ln (\frac{x^{2}+1}{x^{2}+3})+C$
$f(3)=\frac{1}{2}(\ln 10-\ln 12)+C$
$\Rightarrow C=0$
$f(4)=\frac{1}{2} \ln (\frac{17}{19})$
