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Problem 15

Finding an Indefinite Integral In Exercises 15- 36 , find the indefinite integral and check the result by differentiation.

$\int(x+7) d x$

Answer

$$

\frac{x^{2}}{2}+7 x+C

$$

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## Discussion

## Video Transcript

Welcome back, everybody. Today we're gonna take our intervals to the next level by putting two or more together. In this case, we're gonna take a look at our into rules. They pulled him up here. Our integration rules and what we're gonna be doing is we're going to be using number two right here, which says the sum the integral the summer to function is equal to the sum of their integral sze what that means and how it could apply it. If we take the question the integral of explosive in D X, this is gonna be equal to the integral of Axe DX added to the INS agro of seven d X Now you won't have to write this every single time. A separate integral Sze what for our starting purposes, it's very helpful to see it this way. Now, in the first case Okay, we have the integral of acts raised to the first power. Okay, now doesn't matter what variable is used here. You could see from number four which we learn previously. Then when we take the integral off a base race to a power what we do as we raised the exponents by one. So one plus one is two over one plus one is two. So that becomes ax squared over chip. And again, we're going to have to remember that Constant plus c. We'll put her to see one right now. The Nets. This one requires us to d'oh the integral of seven D X. Now you may want to pull the seven out to the front, okay? Or you could just leave it as it ISS if you notice. We're referring now to number three number three. Okay. Which is saying that the integral of really one d'you if you could think of it as axe to the zero. Right? So it was really excellent Zero because Exodus zero was one. You really getting acts to the zero plus one over zero plus one which would become seven s plus or a second constant? Okay, at this point, remember that this whole expression okay, can be put together. The C one and the sea to is really one constant. Okay, so we get axe to about that ex Chu x squared goes, Come, come not behaving, not behaving X squared. There we go over to plus seven x plus c is our answer. Okay? Huh? Wait till I don't have to use this mouse. Okay. Numbering to box or answers this you could also write. Remember, as 1/2 X squared. Now, this constant is very, very important. Okay, because it represents an array off off functions that will eventually will show you have the same shape with regards slow fields that comes later in the course over. Right now, this is the basic integration technique. Now, how do we know if we are correct? What we do is we take our answer, and now we integrate. Sorry, we differentiate. So there, opposite operations here are differentiation rules. Okay, so at this point, we have 1/2 x squared. Okay? If we're gonna find the integral Sorry, the derivative of this, which is D. D s, the d s. Okay. What we're going to do is, as we learned from our integration are differentiation rules. Excuse me. Which is right here are a simple power rule. Okay, Is this Exponents comes to the front, multiplies the constant two times 1/2 is one. So one acts to the first and so is reduced by a power of one. Okay, our second piece is if we differentiate so D. D s. That's another way. That's like the verb former saying Take the derivative dds off seven X is and that's our constant multiple rule. Okay, so it's right here. Constant multiple rule. I remember there's a one year. Okay, so if you ever get confused, it's one multiplied by seven, which is seven X and then reduce this by power of 11 minus one is zero. And as we said before, X zero is 17 multiple by one is seven. So if we put it all together, we just shocked this function that we got by differentiating to get back our original which Waas X plus seven. So we were able in this lesson two. Take the Inter go off the sum of two functions by finding the sun they're into girls And then we're able to check our answer by differentiation. I hope you enjoy this lesson and I'll see you at the next one

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