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Find the derivative of the function.

$ y = \displaystyle \int^{\sin x}_{\cos x} \ln (1 + 2v) \, dv $

$\sin x \ln (1+2 \cos x)+\cos x \ln (1+2 \sin x)$

03:10

Frank L.

00:58

Amrita B.

Calculus 1 / AB

Chapter 5

Integrals

Section 3

The Fundamental Theorem of Calculus

Integration

Sam K.

May 25, 2020

do you not need to find the antiderivative of ln(1+2v) ?

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Okay, We're giving this inter girl and we want to find the derivative of it. So what we're gonna use is the second part of the fundamental therm of calculus that says when you take the derivative of the integral from some constant toe x of some function f of T d t, you get f of X. Okay, so when you take the derivative from a number tow x of a function, you get that function with X plucked in. Okay, So first of all, we have to functions here Cosine X and sign. Next, we're gonna have to split it up. One is on the bottoms. We're gonna have Thio take care of that, and neither one of them are X. But it's okay, So here we go. Let's break it up into into girls. Let's make a cosine x 20 of the l N of one plus, uh, to v TV, plus the integral zero to Synnex Ln one plus two fee TV Can you know we have to turn this one upside down because the constant has to be at the bottom and the function on the top to get this fundamental theorem toe work right. So if you switch the order here, that makes it negative. So we have zero co Synnex Ln one plus Tuffey Devi plus zero to Synnex Ln one plus two V D v. Alright. So here's how this is gonna work. You're gonna take the derivative. So what you're going to get is the function with the cosign plugged in times the derivative of the co sign. Okay, It's the chain rule can first got to tell you I'm taking the derivative, the derivative with respect to X of Why is minus okay? The derivative of the integral is the function with the top thing plugged in. Yeah, times the derivative of the top thing and the derivative of the co sign. Negative. This sign. Okay, Same thing here. The drift of of the integral is the function. So I forgot the one in there Ln with this thing plugged in times the derivative of this thing and the derivative of the sign is the co sign. Okay. And then you could rewrite it if you want, because you have a negative and a negative. And you could put the sign out in the front. So you get sine X Elena one plus two calls my next plus goes on x times the l N of one plus two sine x So remember to use this part of the fundamental the're, um you have to have a constant on the bottom, Okay, zero or whatever. And then the function is on the top. And then when you take the drift of you, get this with this plugged in times the derivative of this, whatever it ISS.

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