Definite integral which i encountered in an examination

I encountered the following definite integral in an examination

$$\int_\pi^{2\pi}\biggl(\frac{x^2+2}{x^3}\biggl)\cos x \,dx$$

To this day i haven't figured out how to evaluate this. I've tried everything I know. Any hints on how to proceed will be highly appreciated.

• Break the integral into two terms about plus and then use IBP for each term. – Dhruv Kohli - expiTTp1z0 Jul 9 '17 at 6:17
• This is an interesting example. It seems that $c=2$ is the only constant for which $f(x) = \left(\frac{x^2+c}{x^3}\right)\cos x$ has an elementary antiderivative. – MathematicsStudent1122 Jul 9 '17 at 6:34
• @MathematicsStudent1122. You are right ! Otherwise, we would see cosine integrals appearing. Thanks for pointing it. – Claude Leibovici Jul 9 '17 at 6:45

Hint. Note that by integrating by parts $$\int\frac{2\cos x}{x^3} dx= -\frac{\cos x}{x^2}- \int \frac{\sin x}{x^2} dx= -\frac{\cos x}{x^2}+ \frac{\sin x}{x}-\int \frac{\cos x}{x} dx.$$