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Question Number 128542 by bramlexs22 last updated on 08/Jan/21

$$\int \frac{{({\mathrm{x}}^4-\mathrm{x})}^{1/4}}{{\mathrm{x}}^5}\mathrm{dx}=?$$

Answered by liberty last updated on 08/Jan/21

$$\gamma =\int \frac{\mathrm{x}{(1-{\mathrm{x}}^{-3})}^{1/4}}{{\mathrm{x}}^5}\mathrm{dx}$$$$\gamma =\int {\mathrm{x}}^{-4}{(1-{\mathrm{x}}^{-3})}^{1/4}\mathrm{dx}$$$$\mathrm{let}\mathrm{z}=1-{\mathrm{x}}^{-3}\Rightarrow \mathrm{dz}={3\mathrm{x}}^{-4}\mathrm{dx}$$$${\mathrm{x}}^{-4}\mathrm{dx}=\frac{1}{3}\mathrm{dz};\gamma =\frac{1}{3}\int {\mathrm{z}}^{1/4}\mathrm{dz}$$$$\gamma =\frac{4}{15}{(1-{\mathrm{x}}^{-3})}^{5/4}+\mathrm{C}=\frac{4}{15}{\left(\frac{{\mathrm{x}}^3-1}{{\mathrm{x}}^3}\right)}^{5/4}+\mathrm{c}$$

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