How to Integrate Using U-Substitution (NancyPi)

MIT grad shows how to do integration using u-substitution (Calculus). To skip ahead: 1) for a BASIC example where your du gives you exactly the expression you need in order to substitute, skip to time 1:30. 2) For an example where you have to REARRANGE THE DU by multiplying or dividing because the du has a different number or sign than what you need, skip to time 8:21. 3) For one where you have to REARRANGE THE U by subtracting or adding because the du expression cannot give you the expression you need, skip to 17:15. 4) For u-substitution with TRIG (SIN/COS) and the power rule, skip to 22:35. Nancy formerly of MathBFF explains the steps.

With all u-substitution integration problems:

The FIRST STEP is to pick your "u". The best choice is usually the longer x-expression that is inside a power or a square root or the denominator, etc (in an "inside function"). Set u equal to this x-expression.

The SECOND STEP is to find "du" by taking the derivative of the u expression with respect to x. For instance, if you have u=3x+2, your du would then be du=3dx. **NOTE: Remember to include "dx" at the end of your du differential expression.

The THIRD STEP is to substitute u and du into the integral everywhere in place of x and dx. **NOTE: if your du does not perfectly match what you need in order to completely substitute before integrating, you must rearrange the du, or sometimes rearrange the u, in order to fully substitute before integrating. For an example of each, see example #2 (time: 8:21) and example #3 (time: 17:15).

The FOURTH STEP is to integrate, remembering to add "+ C" at the end since you integrated an indefinite integral (no limits). The du goes away when you integrate.

The LAST STEP is to "back-substitute" by replacing everywhere u appears with the x-expression that you chose u to be.

[melissag.]

Does she have anywhere where we can donate to her for providing these wonderful videos? Made this so easy to understand and was way more helpful than my professor. Thanks so much for making such quality videos and going step-by-step. It makes those of us who are taking classes a decade after we finished high school much easier to understand! If you have anywhere I can donate to you for your time/help I would love to know!

[androidandroid]

The best thing about Nancy is that she never, never, goes into full math professor mode and says: "and it is obvious that...". She understands that if you are looking at these videos, there is nothing obvious about math. She does not skip three steps or assumes that you will figure it out. By far, one of the best teachers on this subject on line. Hope she will post more (she probably graduated already and is pulling mid 6 figures at some super corporation).

[karlesmcquade]

I've studied calculus and I'm a teacher. These videos on calculus are easily the best series on the topic I've ever seen. Nancy relates easily to the real struggles of calculus students, each step in the approach connects clearly to the previous one, and her set-up of the clear board looks awesome but is very effective and non-distracting. Beautiful work.

[monicam.worden]

Nancy! Fewer tears have been shed after I discovered your videos. I'm beyond grateful.

[bigseekersb]

I have retired form full-time work and want to be a math tutor, to work part-time. I took calculus over 20 years ago and I am relearning it. Nancy, your vids are crystal clear. You are slow and calm, and I love that. Great work! Thanks!

[solidus]

When she explains something complex, suddenly it becomes simple. Teachers around the world need to take this girl as an example.
Please keep up the amazing work! You got my support until the end!

[itsabelleee]

Took Calculus 1 two years ago and Pre-Calculus nine years before that. I'm in Calculus 2 now, and I had no idea there were such things as a "refresher course." Needless to say I've been trying to re-teach myself Trigonometry/Precalculus in order to relearn Calculus 1 in order to even grasp Calculus 2. My professor has been explicitly asked twice over the past few weeks, "How do you know what to take as u?" and his answer was the same each time: "Think of it as... an art. You can't really explain. It's uhhh, intuitive." (What an educator.) Sigh. These past 3 weeks have been rough. Like, really rough. Thank you for softening this hard life♥︎ You (and Professor Leonard) are incredible at translating this crazy conceptual language that is mathematics into sweet plain English. Thank you thank you thank youuu. My first exam is in a couple days and it's a sweet feeling knowing I might actually pass 😭

[drewpetersen]

I really like how you provide the different "types" of problems you will see when using this process. I know you do that with all of your videos, and it's really helpful for us to learn that, even if we are practically copying when we see it in application, but it makes it easy to catch on how it works in general later. In other words, the way you make your videos is probably the most helpful from what else I've seen on YouTube. Thanks!

[SpitKid]

i have so much respect for this girl for being able to write everything in this entire video backwards just so we can see it the right way.

[jhilikjhinukb]

Nancy, not only were you a great source of clear information and support during fairly trying times-the pandemic has pretty much wrecked our schooling system-but you're also a great source of inspiration and clear-headedness for all of thus. Thank you so much for you invaluable content.

[ZippyCoyote]

Thanks to you and your old channel, I passed Calculus 2 so I don't really need to watch this video anymore. But I still wanted to give you the recognition, views, and likes that you deserve! Please keep the videos up. You are helping so many of us struggling students!

[zanedavidson]

Usually, I get distracted by the teacher if they look this good, but this channel has the opposite effect on me and helps me stay engaged and listening. I've come to enjoy topics I normally wouldn't have. Thank you so much!!

[danielcervini]

WONDERFUL !! I just went through this entire video very, very slowly, making sure I understood absolutely everything that was written. And it was almost fun. I think this is as painless learning how to integrate using u-substitution will ever get. Thank you. Thank you. Thank you.

[swindler]

Nancy, thank you so much for saving my sorry butt again. You got me through calc 2 three years ago, and now you're helping me review for Calc 3, Intro to Linear, and Statistics for ECE. Love your stuff - very clear, very concise, easy to understand. You might just be the only reason I get through this hellscape of a term.

[Bhamilton-wsgo]

I'm extremely impressed with your ability to write everything in a mirror image perfectly

[bicboi]

"Are the videos from the 2006 really helping???, yes YouTube advertisment it is, now leave me alone jesus.

[maryloelr.sanchez]

Out of all the tutorials I've watched? This is one of the most effective ways of teaching and as a student, it helped me the most to understand every detail about the topic that I am looking for. Hoping for the next problem tutorial. Thank you so much.

[tshegofatsomatee]

Honestly thought u-substitution was a difficult and I was quite intimidated by it but you explained it so well and I now have a better understanding of it🥺🥺thank you for this

[brookestephen]

you make reading & writing backwards look easy!
Nice to finally see codified WHEN to use them - Thanks.

[ac]

"What happens when you have a trig function?" Well, I simply cry. However, this video was beneficial.