Inner Product

For the first one, show that <cos(mx), sin(nx)> = 0. Also show that <cos(mx), cos(mx)> = 1 and that <sin(nx), sin(nx)> = 1.

For the second one, you're showing that the "1 norm" is a norm on Rⁿ. Verify that the formula satisfies all the defining properties of a norm: positive definiteness, etc. BTW, this norm simply adds the absolute values of the coordinates of a vector. For example, in R³, ||<2, -1, 3>||₁ = |2| + |-1| + |3| = 6.

 

Integration by parts twice would probably work, or you could look in a table of integrals. Since this isn't about learning to integrate, but is instead and application of integration, it seems reasonable to me to look it up in a table of integrals.

 

Can you elaborate on this a bit more? You haven't said quite enough so that I'm not sure I understand what you're talking about. If you have a given integral on one side, and 1/(s - 1) times the same integral on the other side, add -1/(s - 1) times the integral to both sides, and then combine the two integrals using the rules of plain old fractions. At that point you can solve for the integral algebraically.

If that's not what you're talking about, set me straight, please.