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sec-3-x-dx-tan-x-




Question Number 97239 by john santu last updated on 07/Jun/20
∫ ((sec^3 x dx)/( (√(tan x)))) ?
$$\int\:\frac{\mathrm{sec}\:^{\mathrm{3}} {x}\:{dx}}{\:\sqrt{\mathrm{tan}\:{x}}}\:?\: \\ $$
Commented by MJS last updated on 07/Jun/20
t=(√(tan x)) leads to 2∫(√(t^4 +1))dt and we cannot  solve this using elementary calculus
$${t}=\sqrt{\mathrm{tan}\:{x}}\:\mathrm{leads}\:\mathrm{to}\:\mathrm{2}\int\sqrt{{t}^{\mathrm{4}} +\mathrm{1}}{dt}\:\mathrm{and}\:\mathrm{we}\:\mathrm{cannot} \\ $$$$\mathrm{solve}\:\mathrm{this}\:\mathrm{using}\:\mathrm{elementary}\:\mathrm{calculus} \\ $$
Commented by john santu last updated on 07/Jun/20
yes, i can't solve that integral
Commented by MJS last updated on 07/Jun/20
Wolfram Alpha can solve it, I think we can  try to comprehend  ∫(√(x^4 +1))dx=  =(1/3)x(√(x^4 +1))−((2(√i))/3)F (i sinh^(−1)  (x(√i)) ∣ −1) +C  look up the definition of F (ϕ∣m)  sorry I have got no time today
$$\mathrm{Wolfram}\:\mathrm{Alpha}\:\mathrm{can}\:\mathrm{solve}\:\mathrm{it},\:\mathrm{I}\:\mathrm{think}\:\mathrm{we}\:\mathrm{can} \\ $$$$\mathrm{try}\:\mathrm{to}\:\mathrm{comprehend} \\ $$$$\int\sqrt{{x}^{\mathrm{4}} +\mathrm{1}}{dx}= \\ $$$$=\frac{\mathrm{1}}{\mathrm{3}}{x}\sqrt{{x}^{\mathrm{4}} +\mathrm{1}}−\frac{\mathrm{2}\sqrt{\mathrm{i}}}{\mathrm{3}}\mathrm{F}\:\left(\mathrm{i}\:\mathrm{sinh}^{−\mathrm{1}} \:\left({x}\sqrt{\mathrm{i}}\right)\:\mid\:−\mathrm{1}\right)\:+{C} \\ $$$$\mathrm{look}\:\mathrm{up}\:\mathrm{the}\:\mathrm{definition}\:\mathrm{of}\:{F}\:\left(\varphi\mid{m}\right) \\ $$$$\mathrm{sorry}\:\mathrm{I}\:\mathrm{have}\:\mathrm{got}\:\mathrm{no}\:\mathrm{time}\:\mathrm{today} \\ $$

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