$\int_{-\pi}^{\pi}\ x^nf(x) =0$ for all non negative integer n. Then $f$ will be identically $0$.
71071
Nonzero $f \in C([0, 1])$ for which $\int_0^1 f(x)x^n dx = 0$ for all $n$
59911
Looking for help with a proof that n-th derivative of $e^\frac{-1}{x^2} = 0$ for $x=0$.
72660
How can I show that $f$ must be zero if $\int fg$ is always zero?
118873
Prove that $f(x)=0$ for all $x\in\mathbb{R}$.
6990
For $f$ Riemann integrable prove $\lim_{n\to\infty} \int_0^1x^nf(x)dx=0.$
6892
Why is it not true that $\int_0^{\pi} \sin(x)\; dx = 0$?
91304
An entire function whose integral is bounded is identically zero
16212
If $f$ is continuous, nonnegative on $[a, b]$, show that $\int_{a}^{b} f(x) d(x) = 0$ iff $f(x) = 0$
61190
If all derivatives of $f$ are uniformly bounded by a common constant and $f(1/n) = 0$ for all $n$, is $f$ identically zero?
31768
Roots of a functionwith condition $\int_0^\pi f(x) \sin x dx = \int_0^\pi f(x) \cos x dx =0.$
5180
How to show $\alpha\int_a^cf(x)dx+\beta \int_c^b f(x)dx =0$ means $f(x)=0$ everywhere $?$
15886
Proof that $\frac {1} {2\pi i} \oint \frac {{\rm d}z} {P(z)} $ over a closed curve is zero.
11129
Show that $\int_0^ \infty \frac{1}{1+x^n} dx= \frac{ \pi /n}{\sin(\pi /n)}$ , where $n$ is a positive integer.
112323
Showing that $f = 0 $ a.e. if for any measurable set $E$, $\int_E f = 0$
99458
Suppose $\int_{[a,b]}f=0,\text{ then }f(x)=0 \forall x\in[a,b]$
80710
Does $\int_{-1}^1 \frac 1 x dx$ equal zero?
90369
Does $\int_a^bf(x) = 0$ imply $f(x) = 0$ for a Riemann Integral function?