$\int_0^1\bigl( f(x)+f^{-1}(x )\bigr)\>dx=1$
97184
Show: $ f(a) = a,\ f(b) = b \implies \int_a^b \left[ f(x) + f^{-1}(x) \right] \, \mathrm{d}x = b^2 - a^2 $
94869
Show that $ \int_{k}^{2} \frac{1}{f(y)} =1$
122361
Evaluate $\int_0^1\left(\frac{1}{\ln x} + \frac{1}{1-x}\right)^2 \mathrm dx$
71071
Nonzero $f \in C([0, 1])$ for which $\int_0^1 f(x)x^n dx = 0$ for all $n$
5034
A closed form of $\int_0^1\frac{\ln\ln\left({1}/{x}\right)}{x^2-x+1}\mathrm dx$
15409
Calculate $\int_{0}^{1} (x-f(x))^{2016} dx$, given $f(f(x))=x$.
17996
Integral $\int_0^1 \frac{\ln(x+\sqrt{1-x^2})}{x}dx$
37015
Prove $\int_X |f|^p=p\int^{\infty}_{0} t^{p-1}\mu({x: |f(x)>t}) dt\,$
30860
Prove that $f(x)=0$ in $[-1,1]$.
149564
Showing that a function is in $L^1$
131876
Show that $\int_{a}^{b}{x^{n}f(x)dx}=0$, then $f=0$
131692
How to show that $f_{0}f_{1}+f_{1}f_{2}+...+f_{2n-1}f_{2n} = f^{2}_{2n}$
33389
What's the minimum of $\int_0^1 f(x)^2 \: dx$, subject to $\int_0^1 f(x) \: dx = 0, \int_0^1 x f(x) \: dx = 1$?
33989
If $\int_0^1 f(x)x^n \ dx=0$ for every $n$, then $f=0$.
122407
If $f(x) = x^3+3x+4\;,$ Then value of $ \int_{-1}^{1}f(x)dx+\int_{0}^{4}f^{-1}(x)dx$
53519
Finding $\int_0^1{\frac{x^4(1-x)^4}{1+x^2}}dx$
114982
How do I get that $|\int_a^b f| \le \int_a^b |f|$?