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

(B) $\log _{\mathrm{e}} 17-\log _{\mathrm{e}} 18$

(C) $\frac{1}{2}\left(\log _{e} 19-\log _{e} 17\right)$

(D) $\log _{\mathrm{e}} 19-\log _{\mathrm{e}} 20$

JEE Main · 2023 · 25 Jan 2023 · Shift 1 · Indefinite Integrals

Let

f (x) = \int \frac{2 x}{( x ^{2} + 1 ) ( x ^{2} + 3 )} d x

.

If

f (3) = \frac{1}{2} (lo g_{e} 5 - lo g_{e} 6)

, then

f (4)

is equal to

Answer: (A)

With

t = x^{2}

\int \frac{d t}{( t + 1 ) ( t + 3 )} = \frac{1}{2} ln \frac{t + 1}{t + 3} + C = \frac{1}{2} ln \frac{x ^{2} + 1}{x ^{2} + 3} + C

. The condition

f (3) = \frac{1}{2} ln \frac{10}{12} + C = \frac{1}{2} (ln 5 - ln 6)

gives

C = 0

. Hence

f (4) = \frac{1}{2} ln \frac{17}{19} = \frac{1}{2} (lo g_{e} 17 - lo g_{e} 19)

JEE Main · 2023 · 25 Jan 2023 · Shift 1 · Single correct option