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Decide if the function is an antiderivative of $f(x)=2 e^{2 x}$$$F(x)=e^{2 x}+\int_{0}^{1} e^{2 t} d t$$

$F(x)$ is an antiderivate of $f(x)$

Calculus 1 / AB

Calculus 2 / BC

Chapter 6

Antiderivatives and Applications

Section 2

Antiderivatives and the Indef nite Integral

Integrals

Integration

Integration Techniques

Applications of Integration

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Okay, So in this problem, we need to determine whether this function big f of X, which is either the two x plus the definite innit? A girl from +01 of each of the two t d t Is that function an anti derivative of little F of X, which is to eat of the two x and answer. This question is going to be yes. If the derivative of big after backs big F crime of axe is equal to a little f of X because that's what it means for big F of X to be an anti derivative. We're basically just undoing big ffx here with a derivative so that it just becomes little f again. And so all we need to do for this promise we just need to take the derivative of big F Prime Effexor, the derivative big F of X, and see if that gives us little after Becks back. And so the derivative of this termite look complicated. Um, but really, let's just go to turn my terms. The derivative of each of the two X is gonna be too, you know, the two X because we've got this constant in the, um exploded here, so we actually need to bring that down is part of a chain roll. So the derivative the first turn it's going to eat of the two X and then the second term is actually just a number definite ending roles are going to give us numbers. Um, they are not functions. So this is really just a constant. And the derivative of constant, we know is actually zero. So the derivative big F private exes actually just get me to eat of the two X and this is this is equal to little FX. So the answer to this question is, yes, that big FX is an anti derivative of little ffx because big F prime of X.

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