Compute the following:

- where
- where is fixed with and,

To evaluate the integrals of these piecewise defined functions we break the integral into parts so that the intervals of integration correspond to the definition of the functions we have. (We can always split the interval of integration into pieces since the integral is additive with respect to the interval of integration by Theorem 1.17 of Apostol.)

- We split the interval into and to obtain,
- We split the interval into the two intervals and and compute,

sorry, what does fracc^3 means? Thanks

It’s just a LateX error. There’s a missing backslash before “frac”, so the renderer doesn’t know what to do with it. That last term should be (c^3)/(2(1 – c)).