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$\int \frac{1}{\sin\frac{x}{2}\sqrt{\cos^3\frac{x}{2}}}dx$

I tried to solve it.I put $\frac{x}{2}=t$ then $\int \frac{1}{\sin\frac{x}{2}\sqrt{\cos^3\frac{x}{2}}}dx=$

$\int \frac{2 dt}{\sin t\sqrt{\cos^3 t} }=\int \frac{2 dt}{\sin t \cos t\sqrt{\cos t} }$ and then i could not solve.Please help.

diya
  • 3,589

2 Answers2

2

HINT:

Start with $$\dfrac1{\sqrt{\cos\dfrac x2}}=u$$

and write $u^2=1-(1-u^2)$

and use $\#1,\#8$ of this

lab bhattacharjee
  • 274,582
1

$$\int\frac{dx}{\sin\frac{x}{2}\sqrt{\cos^3\frac{x}{2}}}$$ $$=\int\frac{\sin\frac{x}{2}dx}{\sin^2\frac{x}{2}\sqrt{\cos^3\frac{x}{2}}}$$ $$=\int\frac{\sin\frac{x}{2}dx}{\left(1-\cos^2\frac{x}{2}\right)\sqrt{\cos^3\frac{x}{2}}}$$ Let $\cos\frac{x}{2}=t\implies -\frac{1}{2}\sin\frac{x}{2}=dt$ $$=-\frac{1}{2}\int\frac{dt}{\left(1-t^2\right)t^{3/2}}$$

Can you solve further?

Harish Chandra Rajpoot
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