ไม่รู้ว่าโจทย์แบบนี้เคยทำกันหรือยังผมว่าน่าสนใจดีครับ
Evaluate $$\int_{0}^{1}\left(\pi\bigg\lfloor\frac{1}{x}\bigg\rfloor -\bigg\lfloor\frac{\pi}{x}\bigg\rfloor\right) dx$$
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$$\int_{0}^{\frac{\pi}{2}}\frac{a\cos x-b\sin x}{a\sin x+b\cos x}dx=\ln\left(\frac{a}{b}\right)$$
BUT
$$\int_{0}^{\frac{\pi}{2}}\frac{a\cos x+b\sin x}{a\sin x+b\cos x}dx=\frac{\pi ab}{a^{2}+b^{2}}+\frac{a^{2}-b^{2}}{a^{2}+b^{2}}\ln\left(\frac{a}{b}\right)$$