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Question Number 34985 by NECx last updated on 14/May/18

Answered by ajfour last updated on 14/May/18

$$DividingnumeratorandDenominator$$$$by\cos {}^{10}xweget$$$$=\int \frac{\tan {}^{2}x{\left(\sec {}^{2}x\right)}^2}{{(\tan {}^{5}x+\tan {}^{2}x+\tan {}^{3}x+1)}^2}\sec {}^{2}xdx$$$$=\int \frac{t^2{(1+t^2)}^{2}dt}{{(t^5+t^3+t^2+1)}^2}(\mathrm{\forall }t=\tan x)$$$$$$$$=\int \frac{t^{2}dt}{{(1+t^3)}^2}=\frac{1}{3}(-\frac{1}{1+\tan {}^{3}x})+c$$$$So\left(2\right).$$

Commented by NECx last updated on 14/May/18

$$pleasehowdidyoumakethe$$$$substitutionfort?$$

Commented by NECx last updated on 14/May/18

$$Thankyousomuch$$

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