Integral Calculus
Integral Calculus
star_batch_jee_advanced_2025
Grade 12
Question:
Let $f: [1, \infty) \to \mathbb{R}$ and $f(x) = x\int_1^x \frac{e^t}{t} dt - e^x$, then:
$f(x)$ is an increasing function
$\lim_{x \to \infty} f(x) \to \infty$
$f'(x)$ has a maxima at $x = e$
$f(x)$ is a decreasing function
Step-by-Step Solution
Key Concept: Sign analysis of the derivative determines monotonicity of integral-defined functions.
Given $f(x) = \int_1^x \frac{e^t}{t}\,dt - e^x$, differentiating yields $f'(x) = \frac{e^x}{x} - e^x = e^x\left(\frac{1}{x} - 1\right) = e^x\frac{1-x}{x}$. For $x \in [1, \infty)$, since $e^x > 0$ and $1 - x \leq 0$, we have $f'(x) \leq 0$, making $f(x)$ a decreasing function on this interval.
Correct Answer: 3,4