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Question Number 66044 by rajesh4661kumar@gmail.com last updated on 08/Aug/19

Answered by Tanmay chaudhury last updated on 08/Aug/19

∫(dx/(x^5 (1+(1/x^4 ))^(3/4) ))  t=1+(1/x^4 )→dt=((−4)/x^5 )dx  ∫(dt/(−4×t^(3/4) ))  ((−1)/4)×(t^(((−3)/4)+1) /(((−3)/4)+1))+c  (−1)(1+(1/x^4 ))^(1/4) +c

$$\int\frac{{dx}}{{x}^{\mathrm{5}} \left(\mathrm{1}+\frac{\mathrm{1}}{{x}^{\mathrm{4}} }\right)^{\frac{\mathrm{3}}{\mathrm{4}}} } \\ $$$${t}=\mathrm{1}+\frac{\mathrm{1}}{{x}^{\mathrm{4}} }\rightarrow{dt}=\frac{−\mathrm{4}}{{x}^{\mathrm{5}} }{dx} \\ $$$$\int\frac{{dt}}{−\mathrm{4}×{t}^{\frac{\mathrm{3}}{\mathrm{4}}} } \\ $$$$\frac{−\mathrm{1}}{\mathrm{4}}×\frac{{t}^{\frac{−\mathrm{3}}{\mathrm{4}}+\mathrm{1}} }{\frac{−\mathrm{3}}{\mathrm{4}}+\mathrm{1}}+{c} \\ $$$$\left(−\mathrm{1}\right)\left(\mathrm{1}+\frac{\mathrm{1}}{{x}^{\mathrm{4}} }\right)^{\frac{\mathrm{1}}{\mathrm{4}}} +{c} \\ $$

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