$\int_X g \, d\mu =\int_0^\infty\mu(x:g(x)>t) \, dt$
61133
Show that $\int|f(x)|dx=\int_0^\infty m(E_\alpha)d\alpha$
149362
Prove that $ \mathbb{E}[XY] - \mathbb{E}[X] \mathbb{E}[Y] = \int_{- \infty}^\infty \int_{- \infty}^\infty (F(x,y)-F_X(x) F_Y(y)) \,dx\,dy,$
151127
Confirm that $\int_{0}^{\infty}t^{-1}\sin t dt=\pi/2$
10093
$\lim_{x\to \infty}e^{-x^2}\int_x^{x+1/x}e^{t^2}dt$
4656
Prove that there is a $\delta$ such that $\int_{0}^{1} (f(x))^2dx\leq \delta$$\int_{0}^{1} (f'(x))^2dx$ for all $f$ with these conditions
149022
In Markov inequality proof, why is $\int_a^\infty xp(x) \, dx \ge \int_a^\infty ap(x) \, dx$
107987
How do I calculate $u(w)=\int_0^\infty \frac{1-\cos(wt)}{t}\,e^{-t}\,dt$?
120960
$f,g$ such that $\int fg = \int f \int g$
43956
Differentiating $f(x)^{g(x)}$
40019
Showing that $\gamma = -\int_0^{\infty} e^{-t} \log t \,dt$, where $\gamma$ is the Euler-Mascheroni constant.
149399
Given continuous $f$ and $g$, there exists $c \in (a,b)$ such that $\int_{a}^{c} f(x) \ \mathrm dx = \int_{c}^{b} g(x) \ \mathrm dx$
96191
Integral $\int_0^{\infty} \sin(x^2)/x^2\,dx$
67292
Prove that $\lim_{t\to \infty} t\mu(\{x:f(x)\geq t\})=0$
19072
$\int_{\mathbb R} |f(x)| dx < \infty \implies \int^{\infty}_{n} |f(x)| dx \to 0$ as $n \to \infty$?
21339
IF $f(x) = \int_{0}^{\phi (x)} g(t) dt$, How could we find $f'(x)$?
108396
Integral $\int_0^{\infty} \frac{\log x}{\cosh^2x} \ \mathrm{d}x = \log\frac {\pi}4- \gamma$
139742
Bounding the integral $\int_{2}^{x} \frac{\mathrm dt}{\log^{n}{t}}$