Does anyone know how this inner product space question has been solved?

Question

Here is the solution If anyone can explain to me, that'll be great!

Also, is THIS an inner space? For me, it comes out not to be. At the step (u+v,w), it gets disproved.

Answer 1

For your first question, integration by parts is your friend :

$$\int_0^\pi (x+1) \cos(x) dx = \int_0^\pi x \cos(x) dx + \underbrace{\int_0^\pi \cos(x) dx}_{=0} $$

$$= [ - x \sin(x) ]_0^\pi - \int_0^\pi -\sin(x) dx = 0 - [-\cos(x)]_0^\pi = 1-(-1) = 2 $$

For your second question, it's unrelated to the first, but it's clearly an inner product space (forget about the matrix writting, it's the same as the usual $\Bbb{R}^4$ space)