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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分母に2乗根と3乗根の積分

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