I am wondering if there is any closed form expression for $$\int \cos(\theta)^{m} \ \mathrm d\theta,$$ where $m\in\mathbb{R}$. I believe the reduction formula

$$\int \cos(\theta)^{m} \ \mathrm d\theta=\frac{1}{m}\sin(\theta)\cos(\theta)^{m-1} + c + \frac{m-1}{m}\int \cos(\theta)^{m-2} \ \mathrm d\theta$$

only holds for $m\in\mathbb{Q}$. Thus I am wondering if there is a solution by splitting the power into an integer part and a decimal part using the floor function $\lfloor m \rfloor$ as follows: $$ \cos(\theta)^{m}=\cos(\theta)^{(\lfloor m \rfloor+d)}, $$ where $d=m-\lfloor m \rfloor$, which gives $$ \int \cos(\theta)^{m}\mathrm d\theta=\int \cos(\theta)^{\lfloor m \rfloor}\cos(\theta)^{d} \ \mathrm d\theta. $$

Presumably I can use integration by parts on this as long as I choose to integrate $cos(\theta)^{\lfloor m \rfloor}$ since I can use the reduction formula on this?

  • $\begingroup$ If you do not restrict the integration bounds, $\cos \theta$ will have negative values and non-integer powers will be difficult to handle (even if you allow complex values for the integral). $\endgroup$ – gammatester Jan 14 '15 at 9:06
  • $\begingroup$ Thanks @gammatester for your reply. I am only interested in lower and upper limits of integration of the form $L:=((2k+1)\pi)/2$ and $U:=((2k+1)\pi)/2 + \pi$ for $k\in\mathbb{Z}$, which ensures $cos(\theta)$ is positive in these ranges. Forgive my ignorance but why will non-integer powers lead to complex values for the integral? $\endgroup$ – dandar Jan 14 '15 at 13:07
  • $\begingroup$ For example: Using principal values you have $(\cos \pi)^\frac{1}{2} = (-1)^ \frac{1}{2}=\sqrt{-1}= i.$ $\endgroup$ – gammatester Jan 14 '15 at 13:25
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    $\begingroup$ Here another example with your bounds using $k=0$ and $m=\frac{3}{2}.$ Maple gives the following result with the elliptic integral $K:$ $$\int_{\pi/2}^{\pi/2 + \pi} (\cos \theta)^{3/2} d \theta = -\frac{2}{3}i\sqrt{2}K\left(\frac{\sqrt{2}}{2}\right) \approx -1.748 i$$ $\endgroup$ – gammatester Jan 14 '15 at 13:42
  • $\begingroup$ Thanks again @gammatester - very useful examples. However I am still not sure why your second example is complex when $cos(\theta)$ is positive in that range. Can this be shown analytically as to why this is so? Getting back to my original question I presume not because I suspect there is no closed form solution to these integrals when $m$ is not an integer. $\endgroup$ – dandar Jan 14 '15 at 17:10

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