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Give the proper trigonometric substitution and find the transformed integral, but do not integrate.$$\int \sqrt{25+x^{2}} d x$$

$x=5 \tan \theta, 25 \int \sec ^{3} \theta d \theta$

Calculus 1 / AB

Chapter 28

Methods of Integration

Section 8

Integration by Trigonometric Substitution

Integrals

Campbell University

Harvey Mudd College

Baylor University

University of Michigan - Ann Arbor

Lectures

03:09

In mathematics, precalculu…

31:55

In mathematics, a function…

03:14

Integrate using the method…

06:37

Trigonometric substitutio…

05:39

04:55

Trigonometric substitution…

02:48

03:58

09:44

02:11

01:44

So we have this in a girl 25 plus X squared square rooted times dx. And we have to do the appropriate substitution trig substitution so that we would be able to integrate this. So If we use X April two and I can see this 25 here and I know I'm going to be squaring something and I basically want to create a pythagorean identity and we'll find that I know I have the identity of one plus. The tangent squared of an angle is equal to the C. Can't squared of an angle. And I basically want to manipulate to get this type of identity to come up. So I want this coefficient to be five times the tangent update to. That's what I would like my substitution to be. And then notice if I differentiate I would find that dx is equal to five times the C. Can't squared of theta times D. Theta. So now I have my substitution is that I can make not only for X but also for dx. So let's do that. So I have underneath the radical I have 25 plus. And then we're gonna square X. So we're going to end up having 25 tangent squared of theta. And then this dx we need to substitute and will substitute this in place times five. C. Can't square to data times did data. Now we just have to clean it up and do some simplifying. So if I underneath this radical I'm going to pull out that 25 and I have one plus the tangent squared of theta. And that's what I was trying to create. And I have that five C. Can't I swear to data times do fado. Now I can square root this And the square root of the 25 becomes the five. So we have five times. And then this value becomes the square root of she can't square to fada times five. So you can't square two Theta times D. Theta almost there don't step by step. Mhm. And then I can take this five and this five and multiply them together and I can pull the constant out in front. I'll do that. So I had 25 loops And 25. How I wanted to be in black. There we go. So I have 25. Now when I square root this item right here that becomes C. Can of theta. Yeah but I already have this C can't squared of theta. So I end up with C. Can't cubed of state to times D. Theta. So I've done my appropriate substitution with a trig substitution. And that original integral is equal to that in a girl. And we're done we don't have to integrate it.

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