Is all differentiable functions are integrable?

While attending teleconference as a resource person in bangalore. I and panel members was asked to answer the following questions by my fellow lecturers.
Is integrable functions are continous?, Is every continous functions are integrable?, Is all functions are integrable etc.
Really I find these questions are good and we should have to clarify this questions. Here I tried to answer this : students are competent can give your suggestion

-->1. -->Are all functions that can be differentiated, integratable?
All differentiable functions are integrable.
True because all differentiable functions are continuous(Because Differentiability implies continuity but continuity need not implies differentiability) and by FTC, fundamental theorem of integral calculus all continuous functions are integrable.

2.Is every continuous functions are integrable?
True by fundamental theorem of integral calculus

3. Is all integrable functions are continuous?
-->This doesn't follow from the FTC, but I'm having trouble thinking of a counter-example. I looked around on the web and saw a couple people say that this is false, but never explain why. Can you integrate piecewise functions? If so then I can think of an easy counter-example. We've never talked about doing so in class. but think!


Some of the following integrals are not integrable:
LIST OF SUCH INTEGRALS --> --> --> 1. int sinx / x dx

2. int cosx/ x dx

3. int root (sinx) dx

4. int root (cosx) dx

5.int sin (x ² ) dx

6. int cos (x ² ) dx

7. int e ^{-x 2} dx

8. int e ^{x 2} dx

10. int root(1+x ³ ) dx

11. int x tan x dx

12. int 1/ log x dx . and many more ok dear. try to remember it.