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Evaluate the integral.

$ \displaystyle \int_0^1 \sqrt{x^2 + 1}\ dx $

$\frac{\sqrt{2}}{2}+\frac{1}{2} \ln (1+\sqrt{2})$

Integration Techniques

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let's evaluate the integral from zero to one of the square root of X squared plus one. So the expression X squared plus one inside the radical is of the form X square, plus a square so that our trick shop should be of the form X equals eight and data. If a squared is one, then we could just take it to be one so that our troops of his ex people stand data. So this means that the X equal sequence where data and since this is a definite or girl, let's go out and find those new limits of integration the lower limit. So we have X equals zero. So using this equation of here, plug in X equals zero and recall that when we do X equals eight and data the honorable for data is negative power to to power, too. So if tangent equals zero for this interval, the only time that could happen, it's state of equal zero. That's a new lower limit for the upper limit. One. This time we plug in X equals one, so one equals tan Daito. And the only time that happens in this interval appear is that pie before. All right, So these are our new limits of integration in terms of the new variable data. So we have the integral zero power before square root of X squared plus one. So that's ten square. They don't plus one and then DX sequence where? Data debater. So here we can go ahead and simplify this. So we have you think about that night in the town square close. Wanna see cans where so have seek and squared inside the radical. And then we can use the fact that this circled expression here Radical sea cans weird equals C can data. So we have seeking time sequence. Where? Just seeking que bater. I'm running out of room here. Let me go to the next page. So we had zero power before seeking Cute. Now, the trick for this one is to use integration, my parts. So first you can rewrite this here you can go ahead and take you two b c can't So that do you a Sikh and time tangent. And then you could take TV to be what's leftover seek and square so that the equals tan data. So using integration, My part with this choice of U. N B A. We have one half see candidato tan data plus one half natural log seek and data plus Stan data. And then our end points are sea on a pirate for So let's plug goes in, plugging in power For first, we have one half seek and a fiver for his route, too. Tangent of Harbor for is one plus one half natural log seeking of power floors, route, too tangent of power for his one and then plugging in zero. We have one half seeking of zero was one and intention of zero zero plus one half natural log seeking of their of zeros, one tangent of zero zero and then these two terms or zero, and the natural log of one is also zero. So we could ignore those terms. And we could just simplify tohave route to over too, plus one half national log one plus radical, too. And there's our answer