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SL
Numerade Educator

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Problem 4 Easy Difficulty

Evaluate the integral.

$ \displaystyle \int ye^{0.2y} dy $

Answer

$$
25 \cdot(0.2 y-1) \cdot e^{0.2 y}
$$

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SM

Siev M.

August 31, 2020

How did you get Y= 5x?

Video Transcript

Okay. So as we can see, this is, um, indefinite inch girl first after authorization. We saw this exponent exponent. Looks not very nice. So what do we do? Here is, uh, we first apply change of variables. We introduce another variable X and a set X equals to 0.2 times. Why? Just this exponent exponent here. All right? Uh huh. After we do this, we saw why, in terms of X, so why were equal to five effects? Okay. And since this integral involved this dy So we do differential on both sides. So we know dy were echoes too. Five D X. All right, so after we we represent why and d y in terms of X, uh, we plug it in all right, to our formula. Yeah. Uh huh. The integral echoes, too. Five x times e to the power X times five times, DX. Um Then they pull out those constant. This will give us, uh, 25 integral x times e to the power X, the X and this part. We could use a integration by parts. So we first step is we put this exponential function inside the differential, so this equals to 25. Yeah. Integral E x, the u to the Power X right, So that then we use integration by parts that will give us 25 times x times due to the power X minus. We change those, too. I will give us, uh, You change those too, So this will become into the power x integral. All right, then. The next step is to just, uh we find the anti derivative for E to the Power X, which is itself. So let's so so the next page. All right, so the result will be 25 times x times into the power X minus into the Perec's. All right. Oh. Then we recover our original variable. Why? What we do remember our x echo 20 going to why? Yeah. So this, uh, substitute X into the formula will get This is an indefinite integral echoes to 25 times 0.2. Why? Minus squad times into the power 0.2. Why? Yeah, and this will be our answer