Day26: The Daily Math Challenge 2025-12-10 11:40 | Math | 23 | larwar 78 Words | A minute $$\int \frac{\ln (e^x-1)-x}{e^x-1} dx$$ $$\int \frac{\cos ^2 x}{\sin x + \sqrt{3} \cos x} dx$$ 设$f(x)$在$(-\infty, +\infty)$上连续，在$x=0$处可导，且满足$$f(x)=x^3+x^2\lim_{x \to 0} \frac{f(x)}{x}-x\int_0^1 f(x) dx.$$ 求函数$f(x)$的表达式。
$$\frac{1}{2} ln^{2} |1-\frac{1}{e^{x}}| + C$$
$$\frac{1}{8} ln \frac{1+cos(x+\frac{\pi}{3})}{sin(x+\frac{\pi}{3})} + \frac{1}{2} sin(x + \frac{\pi}{6}) + C$$
$$f(x) = x^{3} – \frac{3}{22}(x^{2}+x)$$