Convergence $\int_0^{\pi/2}\frac{e^{\alpha \cos x}-\sqrt{1+2\cos x}}{\sqrt{\cos^5x}}\mathrm{d}x$

Question

Check if the following integral converges:

$$\int_0^{\pi/2}\frac{e^{\alpha \cos x}-\sqrt{1+2\cos x}}{\sqrt{\cos^5x}}\mathrm{d}x$$

So, the point at we should study the integrand is $\pi/2$, but how to do this? This function is so mixed…

Answer 1

If you compute the first three terms of the Taylor series of the numerator of the expression that you are integrating, you will get that\begin{multline}e^{\alpha\cos(x)}-\sqrt{1+2\cos(x)}=\\=(1-\alpha)\left(x-\frac\pi2\right)+(1+\alpha^2)\left(x-\frac\pi2\right)^2+O\left(\left(x-\frac\pi2\right)^3\right),\end{multline}whereas the denominator is$$\sqrt{\cos^5(x)}=\sqrt{-\sin^5\left(x-\frac\pi2\right)}\simeq\left(x-\frac\pi2\right)^{5/2}$$So, near $\frac\pi2$, if $\alpha\ne1$, then your function behaves as $\left(x-\frac\pi2\right)^{-3/2}$ and therefore your integral diverges. But, if $\alpha=1$, then, again near $\frac\pi2$, your function behaves as $\left(x-\frac\pi2\right)^{-1/2}$ and therefore your integral converges.