I have discovered that question 5 below could also be solved using trig substitution rather than Partial Fractions. Take your pick.

Don't have my book or audio lectures? You can download a sample containing some of these lessons here:

Did you read my tips on how to study and learn Math 1700?  If not, here is a link to those important suggestions:

Did you see my tips for Assignment 1?

Did you see my tips for Assignment 2?

These are tips for the first assignment in the Distance/Online Math 1700 course, but I strongly recommend that you do this assignment as homework even if you are taking the classroom lecture section of the course.  These assignments are very good (and challenging) practice.  It is possible that you are doing the topics in a different order in the classroom lecture sections, so you may need to wait until later before tackling this assignment.

Here is a link to the actual assignment, in case you don't have it:

Study Lesson 7 (Integrals of Trigonometric Functions), Lesson 8 (Integration by Parts), Lesson 9 (Integrating by Trig Substitution), and Lesson 10 (Integrating Rational Functions) from my Calculus 2 book to prepare for this assignment.

Like my Lesson 7, question 1(a) and 1(b).

Remember what I said about definite integrals in my tips for Assignment 2.  Never let endpoints get in the way.  Focus on solving the indefinite integral first, then return to the endpoints and deal with them.

Like my Lesson 7, question 1. Don't let the even powers full you! This is not a half-angle identity question like the question above! Technically, that cosine in the denominator is a power of -6, and so nothing to do with half-angle identity. Think more in terms of tangent and secant.

Like my Lesson 8, question 1.

Remember what I said about definite integrals in my tips for Assignment 2.  Never let endpoints get in the way.  Focus on solving the indefinite integral first, then return to the endpoints and deal with them.

Like my Lesson 8, question 1.

You may find it helpful to use u sub first. It is not unheard of to use u sub to change an integral into a form that can then be solved by Integration by parts. In cases like that, I use a different variable other than u so as not to confuse it with the u in integration by parts. 

For example, try letting t equal "square root of x", and compute dt.  Do a t substitution, then you should find that the dt integral you created is an integration by parts problem.

Like my Lesson 10, question 1. You can actually use either Trig Substitution or Partial Fractions to solve this one, whichever you prefer. Partial Fractions is better in general because it will create trivial integrals. The work there is in establishing the unknown constants. Trig substitution is perhaps my favourite technique, but the danger there is sometimes the trig integral you create may be too difficult to solve.  For example secant to the power of 5 (sec^5 (theta) d(theta)) is a J-Type Integration by Parts integral that requires a couple of iterations.

But, that is a lesson to learn in general about solving integrals. Sometimes, there are two or three possible methods to consider, and you have to develop the ability to speedily consider the ramifications of a specific method (see 2 or 3 moves ahead in the solution, like a chess player), to see if it is going to lead you down a blind alley or not.

Unfortunately, if you use Partial Fractions, you will only be able to solve one of the unknown constants by cover-up method, leaving you with three other unknown constants to solve.

Rather than solve the unknown constants by subbing in values of t, the technique I illustrate in my book (which would create a three equations with three unknowns problem), this problem works best if you, instead, leave the t values unsubstituted and instead just get rid of the denominators on both sides of the equation and multiply all the terms out on the right hand side. Then, compare like terms. Which is to say, compare the t-cubed term on the left hand side to the t-cubed term on the right hand side to solve an unknown. Then compare the t-squared term on the left to the t-squared term on the right to solve another unknown. Continue in that way, and you should find all the unknowns solve without much fuss.

You will get something like this (this is a similar example, but not identical):Continue like this to set up equations for all the x terms and for all the constant terms (the terms that don't have an x).

Like my Lesson 8, question 1.

Like my Lesson 9, question 1.

Like my Lesson 8, question 1.  Hint: use the half-angle identity for sin^2 or look up the variations of the double-angle identity for cos(2x).

Like my Lesson 10, question 1.

Like my Lesson 5, question 1.  It is just that the integral requires one of these new techniques this assignment has focused on.