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Use the Table of Integrals on Reference Pages $6-10$ to evaluate the integral.$$\int \frac{d x}{x^{2} \sqrt{4 x^{2}+9}}$$

$\frac{-2 \sqrt{4 x^{2}+9}}{18 x}+C$

Calculus 2 / BC

Chapter 6

TECHNIQUES OF INTEGRATION

Section 4

Integration with Tables and Computer Algebra Systems

Integration Techniques

Improper Integrals

Campbell University

Harvey Mudd College

Baylor University

Lectures

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So we have the integral d X over X squared times, the square root of four X squared plus nine. So to get this integral into a form that we can easily integrate, we can use a u substitution. So if we say you equal to two acts, then d U equals, um two D X. And if we noticed from this equation, U equals two acts. X is just equal, um, to you over to. So if we substitute for Ah, you, Andy, you we First, we have to take a 1/2 out of the integral because we're multiplying dx by two in order to get do you So we have d you over x, which is you over to squared times you squared plus square root of you square plus nine on because of you is two x u squared is four x squared. Um, so, first of all, we can do a few things. We can, um, expand this so he gets, um, you over two squared is just gonna be you squared over four because that's a 1/4 in the denominator. We can take the four out to get 4/2 or two outside of the integral d you over you square times square root Have you squared Plus nine. And lucky for us, we have a, um, formula in the table in an integral table that, um, works for inter girls of this form. So the formula goes, um, if you have the integral of d u over you squared times the square root of you squared plus a squared which we do in this case, a squared is nine and a is three that is equal to, um the square root of you squared are negative square it of you squared plus a squared divided by a squared you So if we just plug in the values for this formula, we get two times negative square root you squared plus nine over nine. You and of course, plus C um, from here, Um, all we have to do is just plug in two X for all the use. So we make sure that the integral is back in in, um, form of back in X instead of you. So if we just clear up some space over here, um, this, um, answer becomes negative, too. Times the square root of two X squared or four acts to the second power plus nine over nine times two X, which of course is just 18 x plus. See, and that is your final integral answer.

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