integration of $\int_{}^{}\frac{dx}{\sin^{1/2}x\cos^{7/2}x}$

Question

$$\int_{}^{}\frac{dx}{\sin^{1/2}x\cos^{7/2}x}$$ We got this integral problem here, now fractional powers are causing trouble in substitution. So I tried substituting for both $\sin^{1/2}x$ and $\cos^{7/2}x$ but that is't enough to do this problem. I guess some kind of rearreangement in the terms in required to solve this problem, but I am not able to figure it what and how. Can you give me a clue on this one ?

Answer 1

Hint: Substitute $\tan x=z$. Then the integral becomes $$\int \frac{(1+z^2)}{\sqrt{z}}dz=2\sqrt{z}+\frac{2}{5}z^{5/2}+c=2\sqrt{\tan x}+2/5\tan^{5/2} x+c$$

Answer 2

To add details to the above answer:

$$ \int \frac{1}{\sin^{1/2}(x)\cos^{7/2}(x)} = \frac{1}{\tan^{1/2}(x)\cos^{4}(x)}$$

Now put $\tan(x) = z$ as suggested above and use that $\frac{d}{dx}\tan(x) = sec^{2}(x)$ and the fact that $\sec^{2}(x) = 1 + \tan^{2}(x)$. Then the integral becomes

$$ \int \frac{(1+z^{2})dz}{z^{1/2}}$$

and now proceed as in above example.