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sin-x-tan-x-1-3-dx-




Question Number 212007 by Nadirhashim last updated on 26/Sep/24
    ∫sin(x) ((tan(x)))^(1/3)  .dx
$$\:\:\:\:\int\boldsymbol{{sin}}\left(\boldsymbol{{x}}\right)\:\sqrt[{\mathrm{3}}]{\boldsymbol{{tan}}\left(\boldsymbol{{x}}\right)}\:.\boldsymbol{{dx}} \\ $$
Answered by Ghisom last updated on 26/Sep/24
=∫(cos x)^(−1/3) (sin x)^(4/3) dx=  =(3/7) _2 F_1  ((2/3), (7/6), ((13)/6), sin^2  x) sin^(7/3)  x +C
$$=\int\left(\mathrm{cos}\:{x}\right)^{−\mathrm{1}/\mathrm{3}} \left(\mathrm{sin}\:{x}\right)^{\mathrm{4}/\mathrm{3}} {dx}= \\ $$$$=\frac{\mathrm{3}}{\mathrm{7}}\:_{\mathrm{2}} {F}_{\mathrm{1}} \:\left(\frac{\mathrm{2}}{\mathrm{3}},\:\frac{\mathrm{7}}{\mathrm{6}},\:\frac{\mathrm{13}}{\mathrm{6}},\:\mathrm{sin}^{\mathrm{2}} \:{x}\right)\:\mathrm{sin}^{\mathrm{7}/\mathrm{3}} \:{x}\:+{C} \\ $$

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