Basic Integration... How? (NancyPi)

MIT grad shows how to find antiderivatives, or indefinite integrals, using basic integration rules. To skip ahead: 1) For how to integrate a polynomial with the POWER RULE, skip to 1:35. 2) For how to integrate NEGATIVE POWERS of x, FRACTIONAL POWERS of x, and RADICALS/ROOTS, skip to 6:12. 3) For how to integrate x^(-1), or 1/x, with the LOG RULE, skip to 7:23. 4) For examples where you use more algebra to rewrite before integrating, to SEPARATE the numerator of a FRACTION, or EXPAND a PRODUCT in order to use the Power Rule, skip to 10:00. 5) For basic TRIG and EXPONENTIAL examples that use rules from the Table of Integrals, as well as trig identities, skip to 11:36. Nancy formerly of MathBFF explains the steps.

1) POWER RULE: If you're integrating a polynomial, or just a power of x, you can use the Power Rule on each x-term in the polynomial. The Power Rule says that for a term that's just a power of x, such as x^3, you can integrate by raising the power by 1 AND dividing by that number (the new number you got by increasing the power by 1). For example, the integral of x^3 would be (x^4)/4. If there was a constant multiplied in front of the x-power, you can keep the constant and integrate the rest of the term (Constant Multiple Rule). For example, the integral of 6x^2 would be 6 times (x^3)/3, which simplifies to 2x^3. You can repeat these steps to integrate every term in the polynomial and string them together for the full integral. You can also keep the addition and subtraction between the terms (because of the Sum and Difference Rules). NOTE: The anti-derivative of a constant (just a number) is always that number times x (the Constant Rule), so the integral of 1 is 1*x, or x. At the end, it's very important to remember to add a constant of integration, to include a "+ c" at the end of your answer, when it is an indefinite integral (integral with no limits). ANOTHER NOTE: The Power Rule only works when the power is not -1.

2) NEGATIVE & FRACTIONAL POWERS, and RADICALS: For negative powers, you can still use the Power Rule, as long as the power is not -1. For fractional powers, you can also use the Power Rule. When you increase a fractional power by 1, you will have to simplify the power (before integrating) by getting a common denominator. For roots, like the square root of x, it's best to rewrite the radical as a fractional power, and then use the Power Rule.

3) IF THE X POWER IS -1 in the integrand, either written as x^(-1) or 1/x, you have to use a special rule, the LOG RULE, that you can find on a Table of Integrals. It says that the anti-derivative of x^(-1), or (1/x) is the natural log of the absolute value of x, plus c.

4) MORE ALGEBRA: Sometimes you may need to do a bit more algebra before integrating, so that the integrand is in a form that fits a basic integration rule, like the Power Rule. If you're integrating a rational expression, you can sometimes separate the numerator, or break the fraction into separate fractions, simplify each term, and then integrate with the Power Rule. If you have a product of x expressions, you can multiply it out, or distribute the factors so that you have just a polynomial (and can use the Power Rule).

5) TRIG & EXPONENTIAL: You can find a lot of trigonometric (and exponential) integral rules in the Table of Integrals. If you don't see it in the table, you may need to use a trig identity first, to rewrite the integrand into a form that you can integrate.

[daviddial]

Only Nancy could make me watch a video on Integration after she already helped me pass Calculus.

[dsrsk]

Thanks for doing my teacher's job and being the sole reason I can graduate- especially now with COVID. I have a 12-hour difference with my online classes and my teacher often forgets to upload his recorded classes/or make them inaccessible, so all your videos have been a lifesaver.

[justagenosfan]

6:13 my match teacher when I turn my eyes away for 1 second

[brookogden]

Such a great teacher!! I have an A in applied calculus because of you.

Seriously, I really hope she is teaching somewhere so more people can benefit from having an awesome teacher! Thank you Nancy!!

[shafe]

Imagine your math teacher explaining some difficult math stuff with a smile on his/her face

[bradsmith]

No need to apologize for "insulting" with algebra talk. I appreciate that steps and explanations are included. These reminders are helpful sometimes. Another great video . Thanks a lot!

[AvenEngineer]

Hi Nancy!

I've watched a heap of your videos over the past 3/4 years. I just wanted to write a comment to say thanks. I'll be starting my first job as an engineer in a few weeks.

:)

[hiredgoon]

This is so much simpler than how my calculus professor made it out to be. Thank you.
Rewrite. Integrate. Simplify.

[JD-yepw]

She's a tough teacher.
Will not spoon feed you But push you.
These videos motivate me

[Titoelpadre]

Just want to say thank you for your incredibly helpful videos! I literally find myself walking around my campus at 3:00am cramming calculus the day before my exam. ( And actually pulling it off thanks to you! )

[MattKimura]

To me it's important to give small reminders of algebra steps because it's a good reminder for a lot of us who forgot how it's done. I appreciate those reminders, very helpful! The people who end up needing your videos are usually ones with weak algebra skills.

[Invizive]

"You're still alert?"
This is how you startle people in educational videos

[iluvfb]

Really just wanna say your channel is AMAZING. you explain things so beautifully. It makes the hardest things simple.

[Herixxon]

Thank you from the bottom of my heart. <3 Because of your great tutorials, I passed my calculus exam and got my network engineering degree :)

[blessedkutsirayi]

Ohh how I wish I had met Nancy a long time ago. She explains very well and she has patience during her lecture. God bless you, Nancy

[rudhrapanchal]

First, I thought that she was just writing everything backward. And then, I realized that there is something called "mirror the video"...

[itsangelinapeople]

I understood more in 2 minutes than I have in the past two weeks of my class. Thank you, I’ll watch all the ads without skipping.

[endrankluvsdaloko]

This is the first video I've seen of yours, and I had to subscribe pretty much as soon as it started. I love the way you explain things in a way that's not too long and drawn out but that is also clear and helps to explain things. When I was in college, I was going for high school education (math), but to become a high school math teacher, you basically have to become an engineer it seems. I never really got the whole Integrals idea. I was so lost in Calc 3 that I switched to middle school education just so I could be done with taking math courses. I've always loved math and wanted to learn it though so I'll definitely be watching your videos and hopefully finally understanding all that stuff I could never get before.

[RtsFps]

It feels so simple when you explain things step by step. Thank you so, so much

[horlatunde]

this has just helped me wipe away my fears on integration and now i can easily do my test without help nor fear....thanks a lot Nancypi