请教几道数分题

ctchentong

• 设\(f'(x)\in C[0,1]\)，求证：\(\D\int_0^1 x^n f(x)\dif x=\frac{f(1)}{n}+O\left(\frac1n\right)\).
• 设\(f(x)\)在\([0,2\pi]\)上单调，证明：\(\D\lim_{n\to\infty}\int_0^{2\pi}f(x)\sin(nx)\dif x = 0\).
• 设\(f(x)\)在\([0,2\pi]\)上连续，证明：\(\D\lim_{n\to\infty}\int_0^{2\pi}f(x)\sin(nx)\dif x = 0\).