关于666的高级方程式解法

limx+0xlnt(t2+1)(t3+1)dt(0xsintdt)2exp(3k=0x1x+k)x5Li5(1x)x4132x3\lim _{x \rightarrow+\infty}-\frac{\int_{0}^{x} \frac{\ln t}{\left(t^{2}+1\right)\left(t^{3}+1\right)} d t\left(\int_{0}^{x}|\sin t| d t\right)^{2} \exp \left(3 \sum_{k=0}^{x} \frac{1}{x+k}\right)}{x^{5} L i_{5}\left(\frac{1}{x}\right)-x^{4}-\frac{1}{32} x^{3}}

详细解法


第一部分

0lnx(1+x2)(1+x3)dx=0lnx(1+x2)(1+x3)dx01x3lnx(1+x2)(1+x3)dx=0(x+1(x2+1)x2(x3+1))lnxdx0(1x(x2+1)x21(x3+1))lnxdx=37π2432\begin{array}{l}{\int_{0}^{\infty} \frac{\ln x}{\left(1+x^{2}\right)\left(1+x^{3}\right)} d x} \\ {=\int_{0}^{\infty} \frac{\ln x}{\left(1+x^{2}\right)\left(1+x^{3}\right)} d x-\int_{0}^{1} \frac{x^{3} \ln x}{\left(1+x^{2}\right)\left(1+x^{3}\right)} d x} \\ {=\int_{0}\left(\frac{x+1}{\left(x^{2}+1\right)}-\frac{x^{2}}{\left(x^{3}+1\right)}\right) \ln x d x-\int_{0}\left(\frac{1-x}{\left(x^{2}+1\right)}-\frac{x^{2}-1}{\left(x^{3}+1\right)}\right) \ln x d x} \\ {=\frac{37 \pi^{2}}{432}}\end{array}

第二部分

n(n+1)π0xsintdtx2(n+1)xlimn(0xsintdtx)2=4π2\begin{array}{l}{\frac{n}{(n+1) \pi} \leq \frac{\int_{0}^{x} | \sin t d t}{x} \leq \frac{2(n+1)}{x}} \\ {\Rightarrow \lim _{n \rightarrow \infty}\left(\frac{\int_{0}^{x} | \sin t d t}{x}\right)^{2}} \\ {=\frac{4}{\pi^{2}}}\end{array}

第三部分

exp(3k=0x1x+k)=exp(30111+tdt)=8\begin{array}{l}{\exp \left(3 \sum_{k=0}^{x} \frac{1}{x+k}\right)} \\ {=\exp \left(3 \int_{0}^{1} \frac{1}{1+t} d t\right)} \\ {=8}\end{array}

第四部分

Lis(1x)13125x5+11024x4+124333+132x2+1xx5Li5(1x)x4x332x2243+x1024+13125\begin{array}{l}{L i_{s}\left(\frac{1}{x}\right)} \\ {\sim \frac{1}{3125 x^{5}}+\frac{1}{1024 x^{4}}+\frac{1}{2433^{3}}+\frac{1}{32 x^{2}}+\frac{1}{x}} \\ {\Rightarrow x^{5} L i_{5}\left(\frac{1}{x}\right)-x^{4}-\frac{x^{3}}{32}} \\ {\sim \frac{x^{2}}{243}+\frac{x}{1024}+\frac{1}{3125}}\end{array}

整合方程式

37π254lim(0xsintdt)2x2243+x21024+13125\frac{37 \pi^{2}}{54} \lim \frac{\left(\int_{0}^{x}|\sin t| d t\right)^{2}}{\frac{x^{2}}{243}+\frac{x^{2}}{1024}+\frac{1}{3125}}

=37π254limx+(02sintdt)2x)21243+11024x+11025x2=\frac{37 \pi^{2}}{54} \lim _{x \rightarrow+\infty} \frac{\left(\frac{\int_{0}^{2} | \sin t d t )^{2}}{x}\right)^{2}}{\frac{1}{243}+\frac{1}{1024 x}+\frac{1}{1025 x^{2}}}

=37π2544π2243=666\begin{array}{l}{=\frac{37 \pi^{2}}{54} \cdot \frac{4}{\pi^{2}} \cdot 243} \\ {=666}\end{array}

结果


1561966753916.jpg

的结果为:

1561966537085.png

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