Wednesday 27 September class

(After Test 1)

Topics:

$\int \sec(x)\textrm{d}x = \ln\left|\sec(x) + \tan(x)\right| + C$

This is found by a clever trick that lets us use substitution: see here. (Just the first paragraph.) A similar trick will give the indefinite integral of $\csc(x)$. Once we computed these, we add them to our list of integration formulas.

• More integration using Trigonometric substitution: we carried out some indefinite integrals, but because of the short class period there was no time to finish the last step, which is substituting back for $\theta$ in terms of $x$. For some indefinite integrals this is rather straightforward, but not for the two we looked at today.

Before going on to that last step, it might be a good idea to brush up on your right-triangle trigonometry definitions of sine, cosine, tangent, secant, cosecant, and cotangent. We’ll use little right triangles and the Pythagorean Theorem to find the formulas for substituting back in the indefinite integrals.

My notes are unavoidably delayed… I hope this is the last delay! But in the meantime here are some goodies:

I  am linking three examples (four videos, because one of them got cut off before it was over) from PatrickJMT which show trigonometric substitutions similar to my examples. Please view them and pay special attention to the last part when he draws the right triangles and uses them to find the back-substitutions. For each video I’ve indicated the time marker when that part of the process begins.

Unfortunately the last two examples are only available on YouTube and not on PatrickJMT’s website. Sometimes other videos will start autoplaying on YouTube after the linked one finishes. Sorry, I don’t know how to prevent this from happening.

I recommend you view these in the order I have them below, despite being out of numerical order.

Here is an example, from his website: $\int \frac{\textrm{d}x}{\sqrt{9x^{2}+4}}$

Example 2

The use of the right triangle to find the back-substitution starts at about 5:15 on the video.

Here is another example on YouTube: $\int \frac{x^{3}}{\sqrt{16-x^{2}}}\textrm{d}x$

Example 1

The use of the right triangle to find the back-substitution starts at about 14:09 on the video.

Here is another example on YouTube, confusingly also called Example 1 : $\int \frac{x^{3}}{\sqrt{x^{2} + 9}}\textrm{d}x$

The use of the right triangle to find the back-substitution starts at about 11:50 on part 1 and continues to part 2 (which may autoplay when part 1 is finished).

This video is not as polished as his others, which is probably why it didn’t end up on his website.

Homework:

• Study at least the first two of the videos linked above, paying special attention to the use of the right triangles in substituting back after the integration. Make sure that you understand how the right triangle is constructed and used. (You can also read my notes once they are posted of course!)

• Do the WeBWorK: there is one old and one new assignment. For the problems in this new assignment, Trigonometric substitution 2, you are told which substitution to use, so you do not have to guess. The challenge is to use the proper right triangle to find the back-substitution at the end.

• There will be a quiz on Monday: the topic will be Trigonometric integrals of the type that includes powers of tangent and/or secant, and also a definite integral which uses trigonometric substitution like the example I worked last time (or the first problem in Trigonometric substitution 1 on WeBWorK).

Don’t forget, if you get stuck on a problem, you can post a question on Piazza. Make sure to give your question a good subject line and tell us the problem itself – we need this information in order to answer your question. And please only put one problem per posted question!

Monday 25 September class UPDATED

(To be updated – I am typing up notes and quiz solutions and it will take a bit of time.)

Topics:

• More Trigonometric Integrals:

Here are partial notes on the examples I discussed in class (most of them). They prompt you to complete certain parts of the integration, and it will help you prepare for Test 1 if you do this yourself! (Without trying to look up a solution somewhere, that is!)

Here are the relevant videos from PatrickJMT which I have previously linked:

Trignonometric integrals part 3 of 6

(integrals of the form $\int \tan^{m}(x)\sec^{n}(x)\textrm{d}x$)

Trignonometric integrals part 4 of 6

(More examples of $\int \tan^{m}(x)\sec^{n}(x)\textrm{d}x$)

The main thing to remember in these types of integrals is that you want (if possible) to end up using substitution with $u = \tan(x)$, so you want to end up with $\sec^{2}(x)$ in your integrand in order to have $\textrm{d}u = \sec^{2}(x)\textrm{d}x$.

Trignonometric integrals part 5 of 6

(Integrating $\int \sin(mx)\cos(nx)\textrm{d}x$)

In these we make use of the product-to-sum identities, which may be new to most of us!

$\sin(mx)\sin(nx) = \frac{1}{2}\left[\cos(mx-nx) - \cos(mx+nx)\right]$ $\sin(mx)\cos(nx) = \frac{1}{2}\left[\sin(mx-nx) - \sin(mx+nx)\right]$ $\cos(mx)\cos(nx) = \frac{1}{2}\left[\cos(mx-nx) + \cos(mx+nx)\right]$

And you may want to view this one also

Trignonometric integrals part 6 of 6

(More examples of different kinds of trig integrals that don’t fit neatly into the above categories)

• Trigonometric substitution (introduction)

Note to follow…

Homework:

• Study and complete the examples when I post them (or from your notes).

• Do the WeBWorK: In the “Trigonometric Integrals” homework you should at least complete the ones that are powers of sines times powers of cosines by Tuesday night, since that is on Test 1. Also do the new “Trigonometric Substitution 1” assignment, only 2 problems, due by Tuesday night. Do not wait to the last minute!

• Don’t forget that Test 1 is scheduled for Wednesday. See the separate post for more information and review.

Don’t forget, if you get stuck on a problem, you can post a question on Piazza. Make sure to give your question a good subject line and tell us the problem itself – we need this information in order to answer your question. And please only put one problem per posted question!

Test 1 review self-tests, now with answers (UPDATED)

The self-tests are here: answers and some links are below the fold. Please also check Piazza where students may have posted solutions and I may have commented or approved!

MAT1575Test1Review

Please make sure that you have read the course policies as pertains to tests.

Monday 18 September class (Updated)

Topics:

• Some more on integration by parts: Integrating by parts after a substitution; definite integrals that use integration by parts. (Examples 165 and 166)

• Trigonometric integrals of the form $\int \sin^{m}(x)\cos^{n}(x)\textrm{d}x$ with m and n positive integers.

These integrals can become rather long as we see, so keeping track of terms and signs is very important.

Here are some nice videos on trigonometric integrals from PatrickJMT: Highly recommended to view even before we discuss in class!

Trignonometric integrals part 1 of 6

(what we did today, for $\int \sin^{m}(x)\cos^{n}(x)\textrm{d}x$ when at least one of the powers is odd)

Trignonometric integrals part 2 of 6

(what we did today, for $\int \sin^{m}(x)\cos^{n}(x)\textrm{d}x$ when both powers are even)

Trignonometric integrals part 3 of 6

(Similar to what we did today, but for $\int \tan^{m}(x)\sec^{n}(x)\textrm{d}x$ which works a similar way)

Trignonometric integrals part 4 of 6

(More examples of $\int \tan^{m}(x)\sec^{n}(x)\textrm{d}x$)

Trignonometric integrals part 5 of 6

(Integrating $\int \sin(mx)\cos(nx)\textrm{d}x$)

Trignonometric integrals part 6 of 6

(More examples of different kinds of trig integrals that don’t fit neatly into the above categories)

Homework:

• Review the examples discussed in class, and at least the first two videos above.

• Do the WeBWorK:  please do not wait to the last minute! There are two assignments: Integration by Parts 2, and Trigonometric Integrals. You should try at least to do the problems in Trigonometric Integrals which are like the examples we worked in class.

• The quiz on Monday will be on Substitution and Integration by Parts.

• Try to watch the other four videos also, before next time. It will help make the lesson go more smoothly! (I hope)

• Don’t forget that Test 1 is scheduled for the first hour or so of class on Wednesday 27 September. The review self-test and answers will be in a separate post. Please check the course policies on tests.

Don’t forget, if you get stuck on a problem, you can post a question on Piazza. Make sure to give your question a good subject line and tell us the problem itself – we need this information in order to answer your question. And please only put one problem per posted question!

Wednesday 13 September class

Topics:

• Substitution method WeBWorK problem #7

• Integration by parts: polynomial times trig function, polynomial times exponential, exponential times trig function (Examples 159-162 in the textbook)

• Integrating by parts to find antiderivative of $\ln(x)$ (Example 163 in the textbook)

Homework:

• Review and study the examples we discussed in class (listed above).

• Do the WeBWorK:  please do not wait to the last minute!

• I have posted the solutions to the first two quizzes here.

I believe that I have corrected or fixed all of the links in the previous posts on this blog, but if you find a link in any post that is broken or leads to the wrong thing, please let me know!

Don’t forget, if you get stuck on a problem, you can post a question on Piazza. Make sure to give your question a good subject line and tell us the problem itself – we need this information in order to answer your question. And please only put one problem per posted question!

Here is my slideshow on how to use Piazza to best effect: MAT1375:1575-UsingPiazza-slideshow

Monday 11 September class

Topics:

• More on the FTC and problems 16, 17, and 18 of the WeBWorK “FTC part 2” – see also the student questions on Piazza on these problems.

• Integration by substitution.

Here are the examples I discussed in class: I may post slideshows or complete solutions later if I have time

$\int\cos^{3}(x)\sin(x)\textrm{d}x$: here we take $u(x) = \cos(x)$, so $\textrm{d}u = -\sin(x)\textrm{d}x$

The result is $-\frac{\cos^{4}(x)}{4} +C$

$\int x^{2}\sqrt{x^{3}+5)\textrm{d}x$: here we take $u(x) = x^{3} + 5$, so $\textrm{d}u = 3x^{2}\textrm{d}x$

The result is $\frac{2}{9}\left(x^{3}+5\right)^{\frac{3}{2}} +C$

$\int\frac{2x+3}{x^{2}+3x}\textrm{d}x$: here we take $u(x) = x^{2}+3x$, so $\textrm{d}u = (2x+3)\textrm{d}x$

The result is $\ln\left|x^{2}+3x\right| +C$

$\displaystyle \int_{0}^{1} e^{-3x}\textrm{d}x$: here we take $u(x) = -3x$, so $\textrm{d}u = -3\textrm{d}x$: also, don’t forget to change the limits of integration when you substitute!

The result is $\frac{1}{3} - \frac{e^{-3}}{3}$

Homework:

• Last reminder: make sure that you have done everything that is listed on the first day post! Most especially, make sure that your City Tech email address is in the User Information in WeBWorK. There are still a number of student who have not done this, and it means that if you send me an email from inside of WeBWorK I will not be able to reply to your email.

• Review and study the examples we discussed in class (listed above).

• Do the WeBWorK: some is due by 11 PM tomorrow, Tuesday, and please do not wait to the last minute! The homework on the Substitution method is not due until Sunday, but try to do at least 5 problems by tomorrow.

• Tomorrow is primary election day in NY, so if you are a registered voter check out the NYC Campaign Finance Board’s Voter Guide and don’t forget to vote!

I believe that I have corrected or fixed all of the links in the previous posts on this blog, but if you find a link in any post that is broken or leads to the wrong thing, please let me know!

Don’t forget, if you get stuck on a problem, you can post a question on Piazza. Make sure to give your question a good subject line and tell us the problem itself – we need this information in order to answer your question. And please only put one problem per posted question!

Here is my slideshow on how to use Piazza to best effect: MAT1375:1575-UsingPiazza-slideshow

Wednesday 6 September class

Apologies for the late post!

Topics:

• The Fundamental Theorem of Calculus part 1 (more)

• Using the FTC together with the Chain Rule

There are three videos from Khan Academy on the FTC linked in the previous post.

Homework:

• Review the examples we discussed in class, and you may also want to view the videos

• Finish the WeBWorK “FTC part 2” – I have delayed the due date until Tuesday because of the lateness of this post.

• There is also a WeBWorK assignment “Substitution Method” but we have not discussed this method yet. You may try some of it if you are interested. We will discuss the Substitution Method next time.

• Make sure that you have done everything from the first day post.

• There will be a quiz at the start of class next time. Be on time! It will be on finding definite integrals.

Don’t forget, if you get stuck on a problem, you can post a question on Piazza. Make sure to give your question a good subject line and tell us the problem itself – we need this information in order to answer your question.

Here is my slideshow on how to use Piazza to best effect: MAT1375:1575-UsingPiazza-slideshow