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Question Number 141691 by mnjuly1970 last updated on 22/May/21

$$....Calculus\left(I\right)....$$$$\mathit{\varphi }:={\int }_{\frac{1}{2}}^1\frac{1}{x^2{(1+x^4)}^{\frac{3}{4}}}dx=???$$

Answered by Dwaipayan Shikari last updated on 22/May/21

$${\int }_{\frac{1}{2}}^1\frac{1}{x^5{(1+\frac{1}{x^4})}^{3/4}}dx$$$$=-\frac{1}{4}{\int }_{\frac{1}{2}}^1\frac{-\frac{4}{x^5}}{{(1+\frac{1}{x^4})}^{3/4}}dx={\left[{(1+\frac{1}{x^4})}^{1/4}\right]}_1^{1/2}=(\sqrt[4]{17}-\sqrt[4]{2})$$

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