How to prove that $ \int_{2}^{x} \frac{dt}{(\log(t))^{k}} = O \Big{(} \frac{x}{(\log(x))^{k}} \Big{)} $ as $x \to \infty$?
139742
Bounding the integral $\int_{2}^{x} \frac{\mathrm dt}{\log^{n}{t}}$
81258
Evaluation of $\int_{-\infty}^\infty \frac{\sin(at)}t\ dt$
27790
Evaluating $\int_0^{\large\frac{\pi}{4}} \log\left( \cos x\right) \, \mathrm{d}x $
20089
Prove that $\lim_{N\rightarrow\infty}(1/N)\sum_{n=1}^N f(nx)=\int_{0}^1f(t)dt$
40564
Prove that $u(x,t)=\int_{-\infty}^{\infty}c(w)e^{-iwx}e^{-kw^2t}dw\rightarrow 0$ if $x\rightarrow \infty$
142408
How to prove Chebyshev's result: $\sum_{p\leq n} \frac{\log p}{p} \sim\log n $ as $n\to\infty$?
151127
Confirm that $\int_{0}^{\infty}t^{-1}\sin t dt=\pi/2$
105707
A proof of $\int_{0}^{1}\left( \frac{\ln t}{1-t}\right)^2\,\mathrm{d}t=\frac{\pi^2}{3}$
76400
Finding $\int^{\infty}_{0}\frac{\ln^2(x)}{(1-x^2)^2} dx$
106812
Find $\int_{0}^{\infty} \frac{\ln(x)}{1+x^2}dx$
104853
Proving that $\lim\limits_{n\to\infty}n\left( \int_0^1 f(t)\, dt -\frac1n\sum_{k=0}^{n-1}f\left(\frac k n\right) \right)=\frac{f(1)-f(0)}{2}$
40537
How to prove $\int_0^\infty e^{-x^2}cos(2bx) dx = \frac{\sqrt{\pi}}{2} e^{-b^2}$
99753
How to prove $\int_0^1\frac{1-x}{(\ln x)(1+x)}\ dx=\ln\left(\frac2{\pi}\right)$?
115089
Definite Integral: $\int_0^1\frac{x^{2} -1}{\log x}\,dx$
147108
Prove that $\int_0^\infty \frac{x\,\log x}{(1+x^2)^2} = 0$
23368
Does $\int_1^2\frac{\sin{x}}{\log{x}} \, dx$ converge?
13747
Proof of $\int_0^\infty \frac{\sin x}{\sqrt{x}}dx=\sqrt{\frac{\pi}{2}}$