分母に2乗根と3乗根の積分
\[
\int\frac{1}{x^{\frac{1}{2}}+x^{\frac{1}{3}}}dx=2x^{\frac{1}{2}}-3x^{\frac{1}{3}}+6x^{\frac{1}{6}}-6\log\left(1+x^{\frac{1}{6}}\right)+C
\]
\begin{align*}
\int\frac{1}{x^{\frac{1}{2}}+x^{\frac{1}{3}}}dx & =\int\frac{1}{\left(x^{\frac{1}{6}}\right)^{3}+\left(x^{\frac{1}{6}}\right)^{2}}dx\\
& =6\int\frac{t^{5}}{t^{3}+t^{2}}dt\qquad,\qquad t=x^{\frac{1}{6}}\\
& =6\int\frac{t^{3}}{t+1}dt\\
& =6\int\left(t^{2}-t+1-\frac{1}{t+1}\right)dt\\
& =2t^{3}-3t^{2}+6t-6\log\left(1+t\right)+C\\
& =2x^{\frac{1}{2}}-3x^{\frac{1}{3}}+6x^{\frac{1}{6}}-6\log\left(1+x^{\frac{1}{6}}\right)+C
\end{align*}
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\[
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\]
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\[
\int_{0}^{1}\left(\sum_{k=1}^{\infty}\frac{\left\lfloor 2^{k}x\right\rfloor }{3^{k}}\right)^{2}dx=?
\]
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\[
\int\sqrt{\tan x}dx=\frac{\sqrt{2}}{4}\log\left(\tan x-\sqrt{2\tan x}+1\right)-\frac{\sqrt{2}}{4}\log\left(\tan x+\sqrt{2\tan x}+1\right)+\frac{\sqrt{2}}{2}\tan^{\bullet}\left(\sqrt{2\tan x}-1\right)+\frac{\sqrt{2}}{2}\tan^{\bullet}\left(\sqrt{2\tan x}+1\right)+C
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\[
\int\frac{1}{\left(1+x^{2}\right)^{2}}dx=\frac{1}{2}\tan^{\bullet}x+\frac{x}{2\left(1+x^{2}\right)}+C
\]