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Evaluate the integral.$$\int \frac{d x}{2 x^{2}+4 x+7}$$

$\frac{1}{\sqrt{10}} \tan ^{-1} \sqrt{\frac{2}{5}}(x+1)+C$

Calculus 1 / AB

Calculus 2 / BC

Chapter 7

PRINCIPLES OF INTEGRAL EVALUATION

Section 4

Trigonometric Substitutions

Integrals

Integration

Integration Techniques

Trig Integrals

Trig Substitution

Campbell University

Harvey Mudd College

Idaho State University

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mhm. Okay, so we have the integral, uh, D X over two x squared plus four x for seven. Yeah. So this is equal Trio the integral E X over two claims X squared, plus ah, two X What's happened? So that is nearly a square. So let's complete it by pulling out a four from the seven. So here's sorry. By pulling out to for so you get two X Plus two her expert plus two x plus one We brought out to into this or brought into this we have to bring out to from there. So you get plus five. Okay, so there is equal to the integral DX, too. Times x squared, plus one square loss for life. Okay. And so that is equal to integral. So if I pull out this one half so I'm calling on the factor of one half. This five becomes five over chill. All right, so now we have one half you enter role X squared plus ones. Where was five halves? All right. And so what does this equal to eso? This looks like in our attention, integral. And the rule for our attention is one of her a where this five house would be a times are contingent time if I perhaps would be a squared, um, times you there are tension of you over a so a and our case is equal to the square root of five over to. So this is equal to one over. That is just the square root of 2/5. We have our factor of one half that we need to bring in. Then we have to York Tianjin's ah, so again we're going to get one over a so speaker room here, one over is going to be Thio five and then you is going to be, uh, X plus one apologies. Serves an error here. Uh, this is X plus one squared. You see if I wrote that again? Yeah. So that should be X plus one squared small mistake, But beak on it. Easy to make mistakes. Okay? And so that's just about done. We need our constancy, But let's simplify this a little bit more. And so home we're going to get here is this is square to this is 1/2. So we're going to get this divine out in this meat. The screwed into. So we'll have one over the square root of 10 or contentions off square roots. Oh, see if we can make that cleaner square root to over five X plus one us, yeah.

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