Question Number 218375 by Nicholas666 last updated on 08/Apr/25
$$ \\ $$$$\:\:\int\frac{\sqrt{\boldsymbol{{tan}}\:\boldsymbol{{x}}}}{\boldsymbol{{sin}}^{\mathrm{3}} \boldsymbol{{x}}\:\boldsymbol{{cos}}\:\boldsymbol{{x}}}\boldsymbol{{dx}} \\ $$$$ \\ $$
Answered by Ghisom last updated on 08/Apr/25
![∫((√(tan x))/(sin^3 x cos x))dx= [t=tan x] =∫(t^(−1/2) +t^(−5/2) )dt=2t^(1/2) −(2/3)t^(−3/2) = =2(√(tan x))−(2/(3(√(tan^3 x))))+C](https://www.tinkutara.com/question/Q218377.png)
$$\int\frac{\sqrt{\mathrm{tan}\:{x}}}{\mathrm{sin}^{\mathrm{3}} \:{x}\:\mathrm{cos}\:{x}}{dx}= \\ $$$$\:\:\:\:\:\left[{t}=\mathrm{tan}\:{x}\right] \\ $$$$=\int\left({t}^{−\mathrm{1}/\mathrm{2}} +{t}^{−\mathrm{5}/\mathrm{2}} \right){dt}=\mathrm{2}{t}^{\mathrm{1}/\mathrm{2}} −\frac{\mathrm{2}}{\mathrm{3}}{t}^{−\mathrm{3}/\mathrm{2}} = \\ $$$$=\mathrm{2}\sqrt{\mathrm{tan}\:{x}}−\frac{\mathrm{2}}{\mathrm{3}\sqrt{\mathrm{tan}^{\mathrm{3}} \:{x}}}+{C} \\ $$