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Evaluate the integrals.$$\int \frac{x^{2}}{\sqrt{x^{2}+9}} d x$$

$\frac{x \sqrt{x^{2}+9}}{2}-\frac{9}{2} \ln \left|\frac{x+\sqrt{x^{2}+9}}{3}\right|+c$

Calculus 1 / AB

Calculus 2 / BC

Chapter 6

Integration Techniques

Section 3

Trigonometric Techniques of Integration

Integrals

Integration

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Okay, so let's start rewriting nine as three to depart to, and then we see that's weaken. Rewrites are ex eyes. Um, X is equal to three tangents of data. And how d X will be able to three seeking square beta. Okay, so let's replace our ex terms. So we get the square or the integral of nine tagine squared of data, and in the X is three took sequins squared data debido over the square root of We have nine and nine here so we can factor are discovered of nine, which is three. And then we have the square root of tendon squared, plus one kind of people to stick its squared of theta so we can get rid rid of Earth Square on our routes. And I wouldn't cancel one of our sequins on or three. And we left with, um, the integral for nine, integral of tangents. Squared of data. I think it's better. You better. Okay, so we want to check the integral falling. So let's see Richard, you to be tangent. We need taken squared, but we don't have that. So if we take you to be Deacon's, okay, we're gonna need a seeking and attention. Okay, I know. What's your addition? In terms of signing coastline. So we have challenges, which is fine. Were they on? That's over coast and square and sequences one over close science It is this coastline cubed. Do you deal And no Um okay, so actually, let's go back a bit If use our identity for his Qianjin squared can rewrite that a sequence. So we commemorate this as, um you can squared of their own minus one and in time chickens data. You think I know we can expand this to get nine Grow up speaking cube better when it's taking a bit off be better and then taking our fallen in a row Me yet nine times, um nine over to kind. Speaking of X, speaking of video tangents they don't and then we have minus natural log of sequins of data plus tangents data and then we have crusty And now let's find our triangles So we have X over three is equal to tangents of it, that is, we have a strangle X and three citizens the square root of X squared Post three squared, which is nine. So what's no that's begins a better. That's people to, um what one of the coastline. Threat. Very over two square root of X squared plus nine and pretend INTs Orbital. We have X over three.

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