Prove that $\int_0^{a}{\int_x^a{t^{-1}f(t)dt}} = \int_0^a{f(x)dx}$
1626
How do we prove $\int_I\int_x^1\frac{1}{t}f(t)\text{ dt}\text{ dx}=\int_If(x)\text{ dx}$
102764
Let $f(x)=\int_0^1|t-x|t~dt$ for all real $x$. Sketch the graph of $f(x)$, what is the minimum value of $f(x)$
67622
Proving of Integral $\int_{0}^{\infty}\frac{e^{-bx}-e^{-ax}}{x}dx = \ln\left(\frac{a}{b}\right)$
79575
The sum of integrals of a function and its inverse: $\int_{0}^{a}f+\int_{0}^{f(a)}f^{-1}=af(a)$
136966
Riemann integral proof $\int^b_a f(x) \, dx>0$
138054
Proving that $\int_0^1 f(x)e^{nx}\,{\rm d}x = 0$ for all $n\in\mathbb{N}_0$ implies $f(x) = 0$
23522
Exponential integral $ \int_0^\infty \frac{x^t}{\Gamma(t+1)}\text dt$
97643
Prove that:$f(f(x)) = x^2 \implies \int_{0}^{1}{(f(x))^2dx} \geq \frac{3}{13}$
109099
Prove that $\lim_{h \to 0}\frac{1}{h}\int_0^h{\cos{\frac{1}{t}}dt} = 0$
105707
A proof of $\int_{0}^{1}\left( \frac{\ln t}{1-t}\right)^2\,\mathrm{d}t=\frac{\pi^2}{3}$
110368
Prove the following integral inequality: $\int_{0}^{1}f(g(x))dx\le\int_{0}^{1}f(x)dx+\int_{0}^{1}g(x)dx$
11557
How to prove $\det(e^A) = e^{\operatorname{tr}(A)}$?
30940
Prove the integral of $f$ is positive if $f ≥ 0$, $f$ continuous at $x_0$ and $f(x_0)>0$
104510
For continuous $f$, $\int_0^x f(t)dt = ∫_x^1f(t)dt$ implies $f=0$.
31319
An integrable and periodic function $f(x)$ satisfies $\int_{0}^{T}f(x)dx=\int_{a}^{a+T}f(x)dx$.
21339
IF $f(x) = \int_{0}^{\phi (x)} g(t) dt$, How could we find $f'(x)$?
61162
Show that $\left(\int_0^1 f(x)\,dx\right)^2\leq2\int_0^1x\, f(x)^2\,dx $