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Give the proper trigonometric substitution and find the transformed integral, but do not integrate.$$\int \frac{d x}{\sqrt{x^{2}-16}}$$

$x=4 \sec \theta, \quad \int \sec \theta d \theta$

Calculus 1 / AB

Chapter 28

Methods of Integration

Section 8

Integration by Trigonometric Substitution

Integrals

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in this question it is being asked that we have to show dr bert Dignam arctic substitution for the home. That is a dx by square root of access square minus 60. Okay, first of all, we know that we can write it as access square minus 16 can be written as forest work. Okay. Now we can see that the form that is here is square root of excess square minus a square. And we know that for this form the better substitution is actually it was two. We have to put here a set theater. Okay, so I'm going to write this substitution here now in your case the value of aces for so we have to make the substitution. That is actually it was 24 sec theater. And now it's a derivative is for into sectors, derivative is sec theater 10 theater and the theater. Okay, so we made this institutions and let I denote this integration by I f So see what happens when we make these substitutions. So I suppose to No matter is one and denominator base, you can see that at the place of X. We're going to substitute for sex data. So it's square and -4 square. And the access replaced by four sector to 10 data and dated. Okay. Now before that simplify it, we can take in the denominator foursquare common. And after detecting it we get here second square theater minus one and multiplied by four said theater 10 theatre and further data. Okay now the identity says that one Palestinian Square Theatre can be written as a square theater. Okay, so I'm going to use this identity here. So you can say that in the denominator regard here, square out of four square is simply fourth and then square out of second squared to minus one is simply 10 square data and four sector to 10 Theatre. The theatre we can cancel out for and after that you can see that square out of 10 theater is simply 10 theta. And now we can also cancel out this 10th. It also and after cancellation of 10 theatre, the simplified form of the integration is I can say that integration of sec theater did it. Okay. This is only asked in this question. It is not told that we have to solve the integration. So you can see this is the simplest form of the integration and to receive this simplest form I had taken the substitution. That is I can say that like physical as to poor said theater. Okay this was my substitution with the help of this substitution. I got the simplest form. Okay, thank you.

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