Question Number 197111 by cortano12 last updated on 08/Sep/23
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Answered by Frix last updated on 08/Sep/23
![∫(dx/( (√(cos^3 x sin^5 x)))) =^(t=tan x) ∫(t^(−(1/2)) +t^(−(5/2)) )dt=2t^(1/2) −(2/3)t^(−(3/2)) =((2(3t^2 −1))/(3t^(3/2) ))= =−((2(1−2sin x)(1+2sin x))/(3(√(cos x sin^3 x))))+C The integral does not converge for x∈[(π/4), (π/2)]](https://www.tinkutara.com/question/Q197121.png)
$$\int\frac{{dx}}{\:\sqrt{\mathrm{cos}^{\mathrm{3}} \:{x}\:\mathrm{sin}^{\mathrm{5}} \:{x}}}\:\overset{{t}=\mathrm{tan}\:{x}} {=} \\ $$$$\int\left({t}^{−\frac{\mathrm{1}}{\mathrm{2}}} +{t}^{−\frac{\mathrm{5}}{\mathrm{2}}} \right){dt}=\mathrm{2}{t}^{\frac{\mathrm{1}}{\mathrm{2}}} −\frac{\mathrm{2}}{\mathrm{3}}{t}^{−\frac{\mathrm{3}}{\mathrm{2}}} =\frac{\mathrm{2}\left(\mathrm{3}{t}^{\mathrm{2}} −\mathrm{1}\right)}{\mathrm{3}{t}^{\frac{\mathrm{3}}{\mathrm{2}}} }= \\ $$$$=−\frac{\mathrm{2}\left(\mathrm{1}−\mathrm{2sin}\:{x}\right)\left(\mathrm{1}+\mathrm{2sin}\:{x}\right)}{\mathrm{3}\sqrt{\mathrm{cos}\:{x}\:\mathrm{sin}^{\mathrm{3}} \:{x}}}+{C} \\ $$$$\mathrm{The}\:\mathrm{integral}\:\mathrm{does}\:\mathrm{not}\:\mathrm{converge}\:\mathrm{for}\:{x}\in\left[\frac{\pi}{\mathrm{4}},\:\frac{\pi}{\mathrm{2}}\right] \\ $$