Indefinite Integration
Partial Fractions / Rational Integration
nta_pyq_2023_jan
Grade 12
Question:
Let $f(x) = \displaystyle\int \dfrac{2x}{(x^2+1)(x^2+3)}\,dx$. If $f(3) = \dfrac{1}{2}(\log_e 5 - \log_e 6)$, then $f(4)$ is equal to
$\dfrac{1}{2}(\log_e 17 - \log_e 19)$
$\log_e 17 - \log_e 18$
$\dfrac{1}{2}(\log_e 19 - \log_e 17)$
$\log_e 19 - \log_e 20$
Step-by-Step Solution
Key Concept: Substitute $t=x^2$; use partial fractions $\frac{1}{(t+1)(t+3)}=\frac{1}{2}\left(\frac{1}{t+1}-\frac{1}{t+3}\right)$ to integrate.
$f(x)=\frac{1}{2}\ln\frac{x^2+1}{x^2+3}+C$. Using $f(3)$: $C=0$. $f(4)=\frac{1}{2}\ln\frac{17}{19}=\frac{1}{2}(\log_e 17-\log_e 19)$.
Correct Answer: 1