1. Remember to take the derivative of what you want to substitute in the integral. Because when we use the method of substitution, we actually change the variable, the whole process changes (the variable and the limits). We need to transfer it completely to get the correct value.
2. If you see a polynomial  under a root sign that you cannot  take its square root directly , think about substituting the variable with some form of trigonometric  functions. There are some characteristics of trigonometric functions that are easy to  be  converted into a form that can be taken square root of,  are favorable for elimination of the variables elsewhere and are easy to differentiate.
3. If it doesn’t work for one way of substitution, don’t quit, try another. For example, if t doesn’t work, try t square; if sin(x) doesn’t work,  try cos(x),etc. It is hard for beginners to foresee if a method could work, because unexpected things happen all the time. Sometimes a variable can be eliminated; and sometimes we think it can be but it actually can’t.

For interval [a,b], we can partition it into n subintervals. And as we take the length of the interval as width, the y value of sample points as height, the area under the curve can be seen as formed by rectangles. As we know, the area of the rectangle can be calculated by taking the product of width and height, and adding up each area together. According to the definition of integral, as we arbitrarily  narrow down the width of each rectangle, ie, the width  of the rectangles arbitrarily approaches to 0 , and the height remains the same, the area for each rectangle will approach to 0.

Though as  the width approaches to 0, the number of the discontinuous points remains the same, which means the number of the rectangles does not change, and the height of the rectangles does not change, only the width goes to 0. Since there are only finite number of rectangles, while the width decreases infinitely, “infinitely” is somewhat powerful than “finite”, thus the total areas of those rectangles contains the “bad points” approaches to 0 as well.