If $\int_x^e f(t)dt = \sin x - \cos x - \frac{x^2}{2}$ for all $x \in R$ then value of $-2f\left(\frac{\pi}{6}\right)$ is _____.
Step-by-Step Solution
Key Concept: Squeeze theorem applied by bounding the sum between two expressions that converge to the same limit.
The function $f(x) = \frac{1}{\sqrt{n^2}} + \frac{1}{\sqrt{n^2+1}} + \frac{1}{\sqrt{n^2+2}} + \ldots + \frac{1}{\sqrt{n^2+2n}}$ is bounded between two Riemann sums. By observing that $\frac{1}{\sqrt{n^2+2n}} < \frac{1}{\sqrt{n^2+1}} < \frac{1}{\sqrt{n^2}}$, we establish that $2 < \lim_{n \to \infty} f(x) < 2$, which forces the limit to equal $2$.
Correct Answer: 1