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Problem 25 Medium Difficulty

Evaluate the integral.

$ \displaystyle \int_0^2 y \sinh y dy $

Answer

$$
\frac{e^{2}+3 e^{-2}}{2}
$$

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Video Transcript

welcome to this lesson in this lesson. Both of the definite integral. Using the integration by parts matter. So integration by part simply haven't two functions that can be expressed as u and DV. And that is he called to do the, uh, the evaluation of I am be the minus the integral from A to B over, do you? Okay, so the best part about this is to identify what is your you and what you see our debate. So here you is a function that is easily differentiable or when differentiated, it goes to zero quickly. Okay, so here will pick that. That's why. So that the you would be called to dy. Okay, Then again, we'll pick the deal. Very as okay. The hyperbolic sign of y y. So that here, if we took the integral e f y is because true they have a bolic costs of y. Okay, so none of that you have, uh, got that. We have the, um we have the way we can divide. Uh, we can find the we'll see why ass i you us? Why, then? May is close. Cool. Okay, so we value that at 02 Will come back to that later. Then I've been becomes then not do you becomes t y. Okay, so here. Yeah, Why then they hyperbolic costs. Now, if we integrate that, we have negative. A parabolic sign of why, Okay, Then you value it to the at this point. Yeah. Now the hyperbolic side why is is given us eat the power y minus e to the negative. Why all over to than the hyperbolic costs? Why is also giving us eat the power plus speed to the power negative y on to. So you will place those mhm then the whole of the differential. Uh, the whole of the integral. Yeah, now becomes, Why then needs the power. Yeah. Yeah, right. Yeah. So that is 02 Then we have this photos for the side. The hyperbolic sine. Yeah. Yeah. Okay, so now this gives us Yeah. If you put the zero out there because of this, it becomes zero. And that is gone. So would have only one by four black. The second part we have right this through. Okay. Mhm. And we have one minus one all over to, uh that is minus one. Okay, so that goes to zero. Well, yeah. I love this. This causes all that. So we have E Yeah, yeah, yeah. Okay. Oh, yeah, Yeah. Mhm. Okay. Yeah, yeah, yeah. Okay, so here. Wow, we have two e Thank you. Last to hope that. Oh, yeah. Not bad. Oh. Oh, Okay. So we're just multiplying to buy this so that we can let all of them sit on a bath. So this becomes very eat about you, then minus e to the power. Negative, too. All over. So this is that and of the lesson. And this is the answer for that definite integral. Yeah. Okay, thanks to a time, The end of the lesson.