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JH
Numerade Educator

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Problem 60 Hard Difficulty

Evaluate the integral.

$ \displaystyle \int \frac{dx}{x^2 \sqrt{4x^2 - 1}} $

Answer

$\int \frac{d x}{x^{2} \sqrt{4 x^{2}-1}}=\frac{\sqrt{4 x^{2}-1}}{x}+C$

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Video Transcript

for this one. Let's start off by just rewriting this So DX we have X squared and then inside let's write that is too X square minus one squared. And the only reason for doing that is to Sir, just unless you have another idea here, we could at least try a trig sub two X equals C can't data. So then here, if you want, you could solve for X before you differentiate So d X equals C can't times tangent all over too. So let's rewrite this inner girl. I thought that one half from over here and then on the top we have the X So that should see Gant time Stan and we already pulled out the two and the bottom We have X squared. So we just square this that sequence where they'd over two square, which is four and then we'll have swear root that seek and square minus one. Now we know that seeking squared minus one equals stands where So when we take the square root we're just left over with him so we can go ahead and cancel this tangent, which is tangent over here. With that tension up there, we have a four that'LL end up in the numerator. If you divided by this to hear you got a two left over, you have a seeking here. You cross it off with one of those your love of the one seeking on the bottom Using the definition of Sikkim. You can write one, oversee Candace co signed data, and we know the integral here is to sign Dana Plus C. We're almost there. We need to get the final answer back in terms of X. So let's draw that triangle based on our strengths of So, here's our triangle. Here's our data. So we have C can data equals two X over one. So that means High Palm News, divided by adjacent is too works over one and then used. Put that really zero to find the other side. And now we could go ahead and evaluate Sign. That's just two times, and then we have opposite overhype avenues and finally here cross off those twos and we have our final answer for X Square, minus one in the radical and then all divided by X and then adding your constant at the very end, and that's your final answer