given $f(-x)=-f(x)$ how to prove that $\int_{-a}^{a} f(x)dx =0 $?
58013
Integral of Even and Odd functions
111515
Proving that $-a=(-1)\cdot a$
21954
A question about $f(x)\equiv 0$
68746
What is $f(f^{-1}(A))$?
30204
Prove that $f'(0)$ exists and $f'(0) = b/(a - 1)$
156235
Are $x \cdot 0 = 0$, $x \cdot 1 = x$, and $-(-x) = x$ axioms?
145573
How to show that $f$ is a zero function?
151519
If $f(f(x))=x^2-x+1$, what is $f(0)$?
116044
Can one prove $\int f > 0$ for $f > 0$ without Lebesgue integration?
90369
Does $\int_a^bf(x) = 0$ imply $f(x) = 0$ for a Riemann Integral function?
136966
Riemann integral proof $\int^b_a f(x) \, dx>0$
30940
Prove the integral of $f$ is positive if $f ≥ 0$, $f$ continuous at $x_0$ and $f(x_0)>0$
16109
Prove that $ \int _a^b\:f\left(x\right)\:\:dx\:=\:\int _a^b\:f\left(a+b-x\right)\:\:dx $
73489
How is the Integral of $\int_a^bf(x)~dx=\int_a^bf(a+b-x)~dx$
13575
Need help for proving that: $f(f^{-1}(A)) ⊆ A$
28949
Prove $f(x) = 0 $for all $x \in [0, \infty)$ when $|f'(x)| \leq |f(x)|$
58647
Show that $f$ is integrable and $\int_{I} f=0$