Indefinite Integration
Partial Fractions with Exponential Substitution
nta_pyq_2023_apr
Grade 12
Question:
Let $I(x)=\displaystyle\int\dfrac{x+1}{x(1+xe^x)^2}\,dx$, $x>0$. If $\lim_{x\to\infty}I(x)=0$, then $I(1)$ is equal to
$\dfrac{e+2}{e+1}-\log_e(e+1)$
$\dfrac{e+1}{e+2}+\log_e(e+1)$
$\dfrac{e+1}{e+2}-\log_e(e+1)$
$\dfrac{e+2}{e+1}+\log_e(e+1)$
Step-by-Step Solution
Key Concept: Substitute $1+xe^x=t$. Then $e^x(x+1)dx=dt$ and decompose $\frac{1}{(t-1)t^2}$ by partial fractions.
$I(1)=\frac{e+2}{e+1}-\ln(e+1)$.
Correct Answer: 1