prove that $\int_{a}^b f \ + \int_{f(a)}^{f(b)} \ f^{-1}=bf(b)-af(a)$
36093
Show rigorously that the sum of integrals of $f$ and of its inverse is $bf(b)-af(a)$
62524
Showing that $\int\limits_{-a}^a \frac{f(x)}{1+e^{x}} \mathrm dx = \int\limits_0^a f(x) \mathrm dx$, when $f$ is even
50655
Proof of $f^{-1}(B_{1}\setminus B_{2}) = f^{-1}(B_{1})\setminus f^{-1}(B_{2})$
45219
Proving that $C$ is a subset of $f^{-1}[f(C)]$
96894
Prove $f_{n+1}^2+f_n^2=f_{2n+1}$
90369
Does $\int_a^bf(x) = 0$ imply $f(x) = 0$ for a Riemann Integral function?
139453
Prove that $\frac{1}{f'(a)} = \frac{1}{2\pi i} \int_C \frac{1}{f(z)-f(a)} dz$
60348
How do I prove $F(a)=F(a^2)?$
76610
Show that $f^{-1}(A \cup B) = f^{-1}(A) \cup f^{-1}(B)$
136966
Riemann integral proof $\int^b_a f(x) \, dx>0$
65923
How to approach proving $f^{-1}(B\setminus C)=A\setminus f^{-1}(C)$?
46844
Show that if $\int_0^x f(y)dy \sim Ax^\alpha$ then $f(x)\sim \alpha Ax^{\alpha -1}$
70512
Prove that $\int f\ d\lambda = \int_{a}^{b} f(x)\ dx,$ for any $f \in \mathcal R[a,b].$
114982
How do I get that $|\int_a^b f| \le \int_a^b |f|$?
30694
Show that there exist $[a,b]\subset [0,1]$, such that $\int_{a}^{b}f(x)dx$ = $\int_{a}^{b}g(x)dx$ = $\frac{1}{2}$
9957
Prove if $f:[a,b] \rightarrow \mathbb{R}$ is integrable, then so is $|f|$ and $\left|\int_{a}^{b}f(x)dx\right| \leq \int_{a}^{b}|f(x)|dx$
18905
$f(a)=f(b)=f_{+}'(a)=f_{-}'(b)=0$ and $|f''(x)|\le M$. Show that $|f(x)|\le \frac{M(b-a)^2}{16}$.