These are all similar to the questions I do in Lesson 4. For the definite integrals given in parts (c) and (d), I recommend you first set up and solve the indefinite integral, then use that solution to compute the answer to the definite
integral.
There is a technique where you can change the endpoints of a definite integral if you have done a u substitution, but I find that unhelpful. Frequently in this course, you will have to use complex methods to solve an integral. Carrying the endpoints of a definite integral just adds to the complications and can lead many students to lose track or express their solutions erroneously.
A far more prudent approach, is, when given a definite integral,
first set up the associated indefinite integral and solve it first, then return to the definite integral and complete your solution.
Part (a)
You must use Polynomial Long Division here! I don't discuss problems like this until Lesson 10 of my book. Study the first few pages of Lesson 10 and look at my questions 1 (a) and (b) for
examples of the approach needed here. Do not study the rest of Lesson 10 though. That will come later.
Parts (b) and (c)
Pretty standard u substitution as taught in Lesson 4.
Part (d)
This integral is not solved using u sub. Whenever there is more than one term on top, always consider splitting the problem up into separate
fractions and then simplifying. Here you can separate the function into x/ sqrt x - 1/ sqrt x. Now, simplify (remember, when dividing the same base, you subtract the exponents).