WEBVTT
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Welcome to our math review video, where we'll look
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at the applications of Integral in physics now for the
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same reason that derivatives are important. Physics integral will
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also be important in physics and will need to be
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able thio operate on the same types of functions.
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If you remember those functions where polynomial you're gonna metric
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functions and exponential functions. If we're able to work
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on these, you'll be able to solve most of
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the problems thrown at you in an introductory physics course
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. Now, um, looking at these, then
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, uh, let's jump right in. If I
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were to take the integral of a of a polynomial
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function, let's just give us a yourselves a generic
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when they x squared plus BX plus c take the
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integral with respect to X here. Then what am
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I going to obtain? Well, uh, remember
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using the power rule in reverse. What we're gonna
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find is Owen. First of all, we should
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distribute thes to make it easier on ourselves so we
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can have think about them one at a time.
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Now, when I take the integral of a constant
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, as I have shown before, what we're going
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to get is a C X. Now there will
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be some additional constant that I'll call see, not
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here because we're using Indefinite. Integral. Generally,
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we won't have that when we're doing applied problems because
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we'll use definite integral. Meanwhile, when I come
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here to be ex well, I know I'm gonna
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have an X squared and my be stays around.
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But what comes here will remember I should be able
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to take the derivative of this and get back BX
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, which means I need a one half because when
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that two falls down, it will cancel out and
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I'll be left with B X all by itself.
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Similarly, I'll have an X cubed here and I'll
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need a one third in front of a. So
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this is going to be my integral of the function
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a X squared plus bx plus c Remember, Integral
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Zahra, little tougher. You gotta think backwards for
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all the rules that you learn with derivatives. Similarly
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, if I'm looking at my trig functions and I
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say okay, I want to take the integral of
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sign of X, the X well, we know
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we're going to end up with cosine. The question
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is, will we have a negative or a positive
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. Now I recommend that you go back and you
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look at the actual plots of this. But if
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you just want a simple rule to recommend the Thio
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to think about, then remember the plot that I
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the picture I wrote before where I had sign of
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x co sign of acts. And when I took
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the derivatives, I came up with co sin of
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X and negative sign of X. Well, in
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this case, what I'm gonna do is I'm going
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to get rid of the arrows going that way and
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instead I want to go backwards. Okay, so
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this will be when I'm applying an integral okay,
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we'll go backwards across. So when I take the
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integral of sine of X, I'll get a cosine
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but this negative still hanging around. So it's going
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to show up here. Meanwhile, when I take
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the integral of cosine of X, I'll just get
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sign of X. I recommend you write this down
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a few times for me having this picture in my
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head. This table is really helpful and has always
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helped me to remember uh, where the negative sign
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goes Now. These will also have constants for the
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same reason as the polynomial. Take a quick look
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, then at Exponential Justus how, before we had
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the derivative of X was equal just to e x
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, We're going to come up with the same thing
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here except, well, I e x plus a
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constant. Now, when we look at this again
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and we say, Well, what if we have
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the integral of e to the X d x?
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Well, we know that the A falls down when
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we get it a derivative, so we'll have to
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pull it down like this so that when we take
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the derivative of this function, we'll get the same
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thing back and we can check that quickly. If
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we were to take the derivative of one over a
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e d d a x plus c not the sea
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not disappears because it's a constant. The A falls
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down, cancels with this A and we're left with
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just e d. A X, which is correct
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. Always remember, if you're worried about whether or
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not you're your anti derivative was correct, just take
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the derivative, and if you get the same function
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you had back then, you have the correct answer