# 2023-07 An oscillatory integral

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
$\left| \int_a^b e^{i f(x)} dx \right| \leq \frac{2}{t}.$

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# 2018-17 Mathematica does not know the answer

For $$a > b > 0$$, find the value of
$\int_0^{\infty} \frac{e^{ax} – e^{bx}}{x(e^{ax}+1)(e^{bx}+1)} dx.$

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# 2016-2 Integral limit

For $$a \geq 0$$, find
$\lim_{n \to \infty} n \int_{-1}^0 \left( x + \frac{x^2}{2} + e^{ax} \right)^n dx.$

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# 2015-16 Complex integral

Evaluate the following integral for $$z \in \mathbb{C}^+$$.

$\frac{1}{2\pi} \int_{-2}^2 \log (z-x) \sqrt{4-x^2} dx.$

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# 2014-14 Integration and integrality

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.)

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# 2013-19 Integral inequality

Suppose that a function $$f:[0, 1] \to (0, \infty)$$ satisfies that
$\int_0^1 f(x) dx = 1.$
Prove the following inequality.
$\left( \int_0^1 |f(x)-1| dx \right)^2 \leq 2 \int_0^1 f(x) \log f(x) dx.$

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# 2012-22 Simple integral

Compute $$\int_0^1 \frac{x^k-1}{\log x}dx$$.

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# 2012-14 Equation with Integration

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 $$t>0$$.

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# 2012-3 Integral

Compute $f(x)= \int_{0}^1 \frac{\log (1- 2t\cos x + t^2) }{t} dt.$

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Let f be a continuous function on [0,1]. Prove that $\lim_{n\to \infty}\int_0^1 \cdots \int_0^1 f(\sqrt[n]{x_1 x_2 \cdots x_n } ) dx_1 dx_2 \cdots dx_n = f(1/e).$