Determining the sign of an integral using the mean value theorem.

Question

Without evaluating the integral , is it possible to prove that $\int_0^{\pi} x \cos xdx$ is negative using the mean value theorem ?

Thanks in advance!

Answer 1

Only a hint, and not using the mean value theorem, but without actually calculating the integral:

Note that $$\int_0^\pi x \cos(x)\, dx = \int_0^{\frac{\pi}{2}} x \cos(x)\, dx + \int_{\frac{\pi}{2}}^\pi x \cos(x)\, dx $$ The first integral on the right hand side has a positive, the second one a negative integrand.

Now transform the second one by mapping the interval $[\pi/2, \pi ]$ affine linearly onto $[0, \pi/2]$ such that $\pi $ mapsto $0$ and $\pi/2$ to $\pi/2$ -- an easy exercise in using the transformation formula. (A reflection across $x=\pi/2$ -- draw a picture). You will be able to the compare the integrands pointwise and conclude your claim.