Suppose that \( f: [a, b] \to \mathbb{R} \) is a smooth, convex function, and there exists a constant \( t>0 \) such that \( f'(x) \geq t \) for all \( x \in (a, b) \). Prove that
\[
\left| \int_a^b e^{i f(x)} dx \right| \leq \frac{2}{t}.
\]
For \( a > b > 0 \), find the value of
\[
\int_0^{\infty} \frac{e^{ax} – e^{bx}}{x(e^{ax}+1)(e^{bx}+1)} dx.
\]
For \( a \geq 0 \), find
\[
\lim_{n \to \infty} n \int_{-1}^0 \left( x + \frac{x^2}{2} + e^{ax} \right)^n dx.
\]
Evaluate the following integral for \( z \in \mathbb{C}^+ \).
\[
\frac{1}{2\pi} \int_{-2}^2 \log (z-x) \sqrt{4-x^2} dx.
\]
Prove or disprove that for all positive integers \(m\) and \(n\), \[ f(m,n)=\frac{2^{3(m+n)-\frac12} }{{\pi}} \int_0^{\pi/2} \sin^{ 2n – \frac12 }\theta \cdot \cos^{2m+\frac12}\theta \, d\theta\] is an integer.
(A typo is fixed on Saturday.)
Suppose that a function \( f:[0, 1] \to (0, \infty) \) satisfies that
\[ \int_0^1 f(x) dx = 1. \]
Prove the following inequality.
\[ \left( \int_0^1 |f(x)-1| dx \right)^2 \leq 2 \int_0^1 f(x) \log f(x) dx. \]
Compute \(\int_0^1 \frac{x^k-1}{\log x}dx\).
Determine all continuous functions \(f:(0,\infty)\to(0,\infty)\) such that \[ \int_t^{t^3} f(x) \, dx = 2\int_1^t f(x)\,dx\] for all \(t>0\).
Compute \[ f(x)= \int_{0}^1 \frac{\log (1- 2t\cos x + t^2) }{t} dt.\]
Let f be a continuous function on [0,1]. Prove that \[ \lim_{n\to \infty}\int_0^1 \cdots \int_0^1 f(\sqrt[n]{x_1 x_2 \cdots x_n } ) dx_1 dx_2 \cdots dx_n = f(1/e).\]