yego.me
💡 Stop wasting time. Read Youtube instead of watch. Download Chrome Extension

Rewriting before integrating | AP Calculus AB | Khan Academy


3m read
·Nov 11, 2024

Let's say that we wanted to take the indefinite integral of ( x^2 \times (3x - 1) , dx ). Pause this video and see if you can evaluate this.

So you might be saying, "Oh, what kind of fancy technique could I use?" But you will see sometimes the fanciest or maybe the least fancy but the best technique is to just simplify this algebraically.

So in this situation, what happens if we distribute this ( x^2 )? Well then, we're going to get a polynomial here within the integral. So this is going to be equal to the integral of ( x^2 \times 3x = 3x^3 ) and then negative 1 times ( x^2 = -x^2 ), and then that times ( dx ).

Now, this is pretty straightforward to evaluate. This is going to be equal to the anti-derivative of ( x^3 ), which is ( \frac{x^4}{4} ). So this is going to be ( \frac{3x^4}{4} ). I could write it that way or let me just write it ( \frac{x^4}{4} ).

Then the anti-derivative of ( x^2 ) is ( \frac{x^3}{3} ), so minus ( \frac{x^3}{3} ). This is an indefinite integral; there might be a constant there, so let me write that down, and we're done.

The big takeaway is you just have to do a little bit of distribution to get a form where it's easy to evaluate the anti-derivative.

Let's do another example. Let's say that we want to take the indefinite integral of ( \frac{x^3 + 3x^2 - 5}{x^2} , dx ). What would this be? Pause the video again and see if you can figure it out.

So once again, your brain might want to try to do some fancy tricks or whatever else, but the main insight here is to realize that you could just simplify it algebraically.

What happens if you just divide each of these terms by ( x^2 )? Well then this thing is going to be equal to, put some parentheses here: ( \frac{x^3}{x^2} = x ), ( \frac{3x^2}{x^2} = 3 ), and then ( \frac{-5}{x^2} ) you could just write that as ( -5x^{-2} ).

So once again, we just need to use the reverse power rule here to take the anti-derivative. This is going to be, let's see, the anti-derivative of ( x ) is ( \frac{x^2}{2} ), ( \frac{x^2}{2} + 3x ), and the anti-derivative of ( -5x^{-2} ).

So we would increment the exponent by 1 (positive 1) and then divide by that value. So there would be ( -5x^{-1} ), we're adding one to negative one, all of that divided by negative one, which is the same.

We could write it like that. Well, these two would just, you'd have a minus and then you're dividing by negative one, so it's really just going—you can rewrite it like this: ( +5x^{-1} ). You could take the derivative of this to verify that it would indeed give you that, and of course we can't forget our ( +C ). Never forget that if you're taking an indefinite integral.

All right, let's just do one more for good measure. Let's say we're taking the indefinite integral of ( \sqrt[3]{x^5} , dx ). Pause the video and see if you can evaluate this.

Try to write a little bit neater: ( \sqrt[3]{x^5} , dx ). Pause the video and try to figure it out.

So here, the realization is, well, if you just rewrite all this as one exponent, so this is equal to the indefinite integral of ( x^{\frac{5}{3}} , dx ). I just rewrote the cube root as the ( \frac{1}{3} ) power ( dx ), which is the same thing as the integral of ( x^{\frac{5}{3}} ).

Many of you might have just gone straight to this step right over here. Then once again, we just have to use the reverse power rule. This is going to be ( x^{\frac{5}{3}} ).

If I raise something to a power and then raise that to a power, I can multiply those two exponents; that's just exponent properties. So, ( x^{\frac{5}{3}} , dx ). We increment this ( \frac{5}{3} ) by 1 or we can add ( \frac{3}{3} ) to it, so it's ( x^{\frac{8}{3}} ).

Then we divide by ( \frac{8}{3} ) or multiply by its reciprocal. So we could just say ( \frac{3}{8} \times x^{\frac{8}{3}} ).

And of course, we have our ( +C ) and verify this. If you use the power rule here, you'd have ( \frac{8}{3} \times \frac{3}{8} ) would just give you a coefficient of 1, and then you decrement this by 1.

You get to ( \frac{5}{3} ), which is exactly what we originally had. So the big takeaway of this video: many times the most powerful integration technique is literally just algebraic simplification first.

More Articles

View All
Does Water Swirl the Other Way in the Southern Hemisphere?
Derek: A couple of years ago my friend Destin and I wanted to definitively answer the question: does water actually swirl the opposite direction down the drain in the other hemisphere? At the time, I was living in Sydney, Australia, and Destin was in Hunt…
Differentiability at a point: algebraic (function isn't differentiable) | Khan Academy
Is the function given below continuous differentiable at x equals 1? They define the function G piecewise right over here, and then they give us a bunch of choices: continuous but not differentiable, differentiable but not continuous, both continuous and …
This morning routine is scientifically proven to make you limitless.
What if I told you that you could transform your life and unlock almost limitless potential, and it only takes about 15 minutes a day? In this video, I’m going to talk about something I’ve been looking for almost all my life: the Holy Grail of morning rou…
Michael Burry Just SHORTED Tesla Stock!
[Music] Hey guys, welcome back to the channel! Got an exciting bit of news to talk about today. Um, controversial, definitely controversial. Michael Burry is shorting Tesla stock. So, Michael Burry, you guys probably know him from “The Big Short.” He’s …
Introduction to photoelectron spectroscopy | AP Chemistry | Khan Academy
In this video, we’re going to introduce ourselves to the idea of photoelectron spectroscopy. It’s a way of analyzing the electron configuration of a sample of a certain type of atom. So what you’ll often see, and you might see something like this on an ex…
"He Saved My Life" American Soldier Returns to Help Iraqi Captain Fleeing ISIS | National Geographic
[Music] [Music] Ian yes for [Music] I’m very scared to lose my son, lose my daughter, lose my wife, thus all my [Music] life. The soldiers, like the captain, are the ones that kept us alive. My name is Chase Msab. I’m a veteran of the Iraq War. I did thre…