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$$\Large\int_{0}^{\pi }\theta \ln\tan\frac{\theta }{2}\mathrm{d}\theta $$ $\Large\mathbf{Solution:}$ 显然 $$\int_{0}^{\pi }\theta \ln\tan\frac{\theta }{ 阅读全文
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1.Table of Integrals,Series and Products , Eighth Edition , I.S.Gradshteyn, I.M.Ryzhik 这就是众所周知的"积分大典",也是最新版,包含了巨量的积分公式,绝对是值得拥有的,遗憾的是只有公式,没有证明,当然,在书的开头 阅读全文
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$$\Large\sum_{n=1}^{\infty} \frac{H_{n}}{2^nn^4}$$ $\Large\mathbf{Solution:}$ Let $$\mathcal{S}=\sum^\infty_{n=1}\frac{H_n}{n^42^n}$$ We first conside 阅读全文
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$$\Large\int_{0}^{1}\frac{\ln^{2}\left ( x \right )\mathrm{Li}_{2}\left ( x \right )}{1 x}\mathrm{d}x= 11\zeta \left ( 5 \right )+6\zeta \left ( 3 \ri 阅读全文
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$$\Large\int_0^1\frac{\ln^3(1+x)\ln x}x\mathrm dx$$ $\Large\mathbf{Solution:}$ Start with integration by parts (IBP) by setting $u=\ln^3(1+x)$ and $\m 阅读全文