Last week we covered trig integrals, trig substitutions and rational integrals. Next week, we will finish with rational integrals and introduce improper integrals. Just to remind you, a test is coming up, so make sure to review!
The major goal for trig substitution is to handle nonlinear functions such as fraction exponents, and typical examples we see are summed up in the table on page 467. By doing these substitutions and using the trig identities, we are left with one whole piece under the square root. Therefore, if you see strange functions under the square roots, try to complete the square and leave only constant outside– an example is Example 7 on page 471. Start making your note sheet and make sure to write down the trig identities on your sheet for your test!
After your trig substitution, you should get an integral shaped like the ones in section 7.2: product of powers of sine and cosine, or product of powers of tangent and secant. Use the general methods we studies in section 7.2 and classify the resulting trig integrals by the powers of the trig functions. The results are summed up in tables on page 462 and 463.
Rational integrals have integrands that are rational functions. Any rational functions can always be simplified to the sum of a polynomial and a proper rational function. By proper, we mean the numerator has degree strictly lower than the degree of denominator. If the rational function is already proper, we do partial fractions and decompose the integrand into things we know the antiderivative of; if not, first simplify by long division.
We covered the case where the results of partial fractions are distinct linear factors, next week we will study the cases where the results of the partial fraction contain repeated factors or irreducible quadratic factors.
The third true and false problem in our test 1 contains an integral that has a undefined spot in the integration domain, such integrals are one kind of improper integrals. Improper integrals are definite integrals that contain either undefined points in the integration interval, or has infinity as one integration limit. For these integrals, we need to be careful and check if it makes sense to compute them first.
Homework problems are in a separate post, make sure to check them out!
Alright, hope you have a good weekend, and don’t forget to do quiz corrections, they are due Nov 12/13!