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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複雑な2重根号を含む定積分
\[
\int_{-\frac{1}{2}}^{\frac{1}{2}}\sqrt{x^{2}+1+\sqrt{x^{4}+x^{2}+1}}dx=?
\]
tanの立方根の積分
\[
\int\sqrt[3]{\tan x}dx=\frac{1}{4}\log\left(\tan^{\frac{4}{3}}x-\tan^{\frac{2}{3}}x+1\right)+\frac{\sqrt{3}}{2}\tan^{\bullet}\left(\frac{2\tan^{\frac{2}{3}}x-1}{\sqrt{3}}\right)-\frac{1}{2}\log\left(\tan^{\frac{2}{3}}x+1\right)+C
\]
sinの3乗をｘの2乗で割った定積分
\[
\int_{0}^{\infty}\frac{\sin^{3}x}{x^{2}}dx=?
\]
気付かないと解けないかも
\[
\int_{0}^{\infty}\frac{1}{\left(1+x\right)\left(a^{2}+\log^{2}x\right)}dx=?
\]