Tips

1. Before you doing a substitution, observe which part of the function is the derivative of the other part of this function.
2. If the function is not so clear that you cannot find out the substitution immediately. Make some tries instead of doing nothing and you may find idea during this process.
3. Remember the derivative of those important function, such as sin or cos, to help you find out the substitution faster and it can also save your time in a exam.

According to the definition of integrable, we divide the area into infinite rectangle and each rectangle have same width. Then, we pick up a f(t*) in each subinterval as the height of each rectangle and calculate each rectangle’s area and sum up.

In this case, there are at least 3 subintervals we can make f(t)=0 (at most 6 if t=1, 2, 3 is at the end point of its subinterval).

When the discontinuous point is finite, we can ignore this different of sum because we take the limit as n growing to infinite and there exist another part of sum which is infinite.