Indefinite Integration
Improper Integral via Partial Fractions
nta_pyq_2023_apr
Grade 12
Question:
$\displaystyle\int_0^\infty\dfrac{6\,dx}{e^{3x}+6e^{2x}+11e^x+6}=$
$\log_e\dfrac{32}{27}$
$\log_e\dfrac{512}{81}$
$\log_e\dfrac{256}{81}$
$\log_e\dfrac{302}{27}$
Step-by-Step Solution
Key Concept: Factor denominator: $(e^x+1)(e^x+2)(e^x+3)$. Partial fractions: $\frac{6}{(e^x+1)(e^x+2)(e^x+3)}=\frac{3}{e^x+1}-\frac{6}{e^x+2}+\frac{3}{e^x+3}$ (after multiplying by $e^x$).
$I=\ln\frac{32}{27}$.
Correct Answer: 1