Definite Integration
Definite Integration
nta_abhyas_2025
Grade 12
Question:
Consider $I(a) = \int_0^a \frac{dx}{x}$ (where $a > 0$), then the value of $\sum_{i=1}^n I(i) + \sum_{i=1}^{n-1} I(i)$ is
$\ln 2$
$1$
$\ln 2$
$\ln 4$
Step-by-Step Solution
Key Concept: Telescoping series where consecutive logarithmic terms cancel systematically
Given $f(a) = \ln a_1^{a_1} - \ln a^2 - \ln a = \ln\left(\frac{a_1}{a^2}\right) = \ln a$. Then $\sum_{i=1}^{5} f(i) = \ln 2 + \ln 3 + \ln 4 + \ln 5 = \ln(2 \cdot 3 \cdot 4 \cdot 5) = \ln(2 + \ln 3 + \ln 4 + \ln 5)$. The sum telescopes and equals zero.
Correct Answer: A