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Find the antiderivative $ F $ of $ f $ that satisfies the given condition. Check your answer by comparing the graphs of $ f $ and $ F $.

$ f(x) = 4 - 3(1 + x^2)^{-1} $, $ \quad F(1) = 0 $

$$

F(x)=-3 \tan ^{-1} x+\frac{3 \pi}{4}

$$

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Campbell University

Harvey Mudd College

Idaho State University

Boston College

Okay, so we're being asked to find the intuitive after satisfies the skin conditions. So, um, let's go ahead. And we're going to take the integral of f of X. So this is gonna be four minus three and this can be rewritten X square plus one dx. And then we can further break this down so we can say four. Integral of DX minus three. Integral. Uh, X square plus one. Oh, Dino, X square plus one is going to be one over X square plus one. And this is D X. So now this is a actually pretty simple integration. So this is this one is going to become excessively four X and then minus three and then one of her X squared plus one. Well, that's just 10 inverse acne and then nobody across Constant. Now, since this is our capital afra, Becks, we can go ahead and plug an f of one so f of one. Well, what does that give us that gives us, uh, four minus, uh three Qianjin Invert, uh x and so and plus C and this is all equal to zero. Um and this is four times one and then this is gonna be four. And in tangent of one, uh, 4, 10 10 Inverse of one is pi over four. There's gonna be three pi over four plus equals zero is all for C. The C comes out to be three pi over four, my, uh, minus four. And then our official anti derivative is going to be four minus 3 10 in verse X plus three pi over four, minus four. And, um, these four actually canceled. So we left the negative 3 10 and verse six plus three point before, and this is our function.