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Question Number 158742 by cortano last updated on 08/Nov/21

Commented by HongKing last updated on 08/Nov/21

$$=\frac{2}{5}{({\mathrm{x}}^6+{\mathrm{x}}^4+\frac{1}{{\mathrm{x}}^4})}^{\frac{5}{4}}+\mathbb{C}$$

Answered by tounghoungko last updated on 08/Nov/21

$$\int \frac{x^5(3x^5+2x^3-2x^{-5})x\sqrt[4]{x^6+x^4+x^{-4}}}{x^6}dx$$$$=\int (3x^5+2x^3-2x^{-5})\sqrt[4]{x^6+x^4+x^{-4}}dx$$$$let\lambda =x^6+x^4+x^{-4}\Rightarrow d\lambda =6x^5+4x^3-4x^{-5}dx$$$$I=\frac{1}{2}\int {\lambda }^{1/4}d\lambda =\frac{2}{5}{\lambda }^{5/4}+c$$$$=\frac{2}{5}\sqrt[4]{{(x^6+x^4+x^{-4})}^5}+c$$$$=\frac{2}{5}\sqrt[4]{\frac{{(x^{10}+x^8+1)}^5}{x^{20}}}+c$$$$=\frac{2\sqrt[4]{{(x^{10}+x^8+1)}^5}}{5x^5}+c$$

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