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Evaluate the integral.

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

$$\ln \left|\frac{\sqrt{1+x^{2}}-1}{x}\right|+\sqrt{1+x^{2}}+C$$

Integration Techniques

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here we have the integral of the square roots of one plus X where all divided by X in the numerator we see an expression inside the radical of the form excluding plus ace cleared In this case, the tricks up his X equals eight and data. And since a equals one, let's take X to be tan data From this it follows that T X is he can swear. And before we start writing the actual answer, let's just simplify this numerator. We have the square root of one plus x squared, so that becomes san squared data. We know that this is equal to the square root of C can't square and that just becomes sick and data so are integral becomes seeking and replaces in radical in the rare sigan data and then the ex sequence where data and in the denominator we just have X, which was tan data. Now we can rewrite this by using a Pythagorean identity for C can square We'LL use the same one we just used a a moment ago. See, can't square is one plus dance where data all over Tan data. What's good and right? This is two fractions we have C can't. Data chance flared over ten. Plus he can't data over ten. Now, For this first expression, we could cancel off one of the hands. Some are just left over with seek and data tan data for the second, we can go ahead and rewrite this as so C can is one over co sign one over tangent is co sign over sign. Yeah, so here you could cancel off those co signs and you get an integral Of course he can. So let's go to the next page. We had C can't date a Pantera, plus Kosi can data. These are both Trigon on girls that we know already in a girl of seek and time. Stan is sick, just chicken. And then for Kosi can we have natural log absolute value? Kosi can tear of minus contention Taito and then plus our constancy at the end. So now we have to write our answer back in terms of X. So going back to our original tricks up ten data equals X, which you can also write as x over one. There's an angle data so of tension of data is X over one We have X here for opposite one for Jason H is our High Palm News. Bye. Petya Grand Terram et Square is exploring plus one. So each is this for root of expert plus one. So now going back to our previous answer, let's go ahead and rewrite this back in terms of X. So C can of data is high pop news over adjacent, so it's just a JJ plus natural log Kosi can is each over X. That's hype out news over opposite. So here's a JJ and then divided backs and then minus coat. Sandra Cho Tension is adjacent over opposite. So one over X and here. The most work we could do is probably just combined these fractions because they have a common denominator plus see, and there's a final answer. Awful.