AP Calculus AB Trigonometric Integrals II

Hello, welcome back to www.educator.com, and welcome back to AP Calculus.0000

Today, we are going to continue our discussion of trigonometric integrals.0004

Let us jump right on in.0007

Let us recall what the integral of tan(x) looks like.0012

Let us work in blue, I really like blue color.0018

Let us recall the integral of tan x dx.0022

I'm going to rewrite that as the integral of sin x/ cos x dx.0032

And then, I'm going to let u equal cos x and du is equal to -sin x dx.0040

Sin x dx is equal to –du.0052

What we end up actually getting is, this thing is going to be the integral of -du/u.0059

Let us come over here = -the integral of du/u = -natlog of u + c is equal – natlog.0071

We set that u was cos x = x + c.0085

The integral of the tan is –natlog of cos x dx.0092

You will also see it written like this, excuse me.0100

This can also be written as the natlog sec x + c.0108

Either one of these is absolutely fine, I personally prefer that one but where this comes from is this.0132

-ln of cos x + c that is the same as 0 - the ln of cos x + c.0141

0 can be written as the natlog of 1.0158

Natlog of 1 - the natlog of cos x + c is equal to the natlog of this over that 1/ cos x + c,0161

which is equal to the natlog of sec x + c.0180

That is where it comes from, either one is fine.0185

The actual one that you go through, gives you this one.0187

This is just another version of it, in case you do not like that negative sign in front, totally a personal choice.0190

We have the integral of the sin, the integral of the cos, and now we have done the integral of the tan which we have done before.0200

Now let us do the integral of secant.0207

Now let us find, and we are doing this just so we actually have a table of the integrals of all of our fundamental trigonometric functions.0211

Now let us find the integral of sec x dx.0223

This one is just manipulation, that is all it is.0228

Again, the manipulation itself is not really important.0232

What is important is that we actually find some expression for the integral of the sec(x).0235

So that we have an integral of a basic trigonometric function, so that we can use it for other integrals,0240

if it happens to come up, which is exactly why we are doing this.0246

It equals the integral of sec x × sec x + tan x/ sec x + tan x dx0249

is equal to the integral of sec² x + tan x/ sec x + tan x dx.0269

We will let u equal to sec x + tan x, du = sec x tan x + sec² x.0282

Sec tan + sec dx/u.0302

This equals the integral of du/u which = the natlog of u + c which = the natlog of sec x + tan x + c.0312

Now we have that one.0331

I'm going to leave the integral of cot x dx and the integral of csc x dx to you.0335

Do the same way that we just did for these two.0355

Now we have the integrals of all six basic trig functions, that is it.0358

Now we can continue our discussion of trigonometric integrals.0379

Let us try this one, let us go to black, finish the integral, the integral of tan⁵ x.0383

Tan⁵ x, I know that tan² x is equal to sec² x – 1.0392

How about if I write this as the integral of the tan(x) × tan⁴ (x) dx?0400

That is going to equal the integral of the tan(x) × tan² (x)² dx.0412

And that is going to equal the integral of tan x × sec² x - 1 dx²,0422

that is equal to the integral of tan x × sec⁴ x – 2 sec² x + 1 dx.0437

That is going to equal the integral first one and then the second one.0453

It is going to equal the integral of tan x sec⁴ x dx - 2 × the integral of tan x sec² x dx + the integral of tan x dx,0461

that is why we did the tan and sec in first because when we simplify some other trigonometric integrals,0483

one of the integrals that is going to show up is the integral of tan x.0489

We wanted to just be able to have something to fill in there as an answer.0492

This is our first integral, our second integral, and our third integral.0497

Our first integral, the integral of tan x sec⁴ x dx, that is going to equal the integral of tan x sec x.0506

I pull out a sec because the sec power is even, I can pull out a sec.0526

That leaves me with sec³ x dx.0531

I’m going to let u equal the sec(x) and I’m going to let du equal sec x tan x,0535

that gives me the integral of u³ du is equal to u⁴/ 4 + c =,0547

Let me do this on the next page, I’m actually going to need that.0571

That actually equals sec⁴ x/ 4 + c, that is our first integral.0575

Our second integral is the integral of tan x sec² x dx.0591

We let u equal tan x, du = sec² dx.0600

This is actually equal to the integral of u du, that is equal to u²/ 2 + c,0607

that is going to be u is tan, it is going to be tan² x/ 2 + c.0617

That is our second integral.0623

Our third integral, that was the integral of tan x dx.0628

We already solved that equals - the natlog of cos x + c.0635

We put them all together.0643

The integral of tan⁵ x dx is going to end up equaling sec⁴ x/ 4.0647

The c’s, I can combine it to 1c, that is not a problem.0668

I have this one, and then + tan² x/ 2.0672

I have this one, - the natlog of the cos(x) + c.0685

There you go, always manipulation, always manipulation.0696

It seems like that is all we do in mathematics, is not it?0701

That is not all we do, that is a big part of it.0703

Now evaluate sec⁵.0713

How do we deal with sec⁵?0717

In the previous lesson, we dealt with even powers of sec, it also has odd power of sec.0720

Let us see what we can do.0728

I'm going to rewrite this as sec⁵ x, sec² x dx.0730

This is how we deal with an odd power of sec, you try something.0748

This is what somebody tried in, it actually ended up working.0754

Let us use integration by parts.0758

Let me work in blue.0769

We will call this u and we will call this dv.0772

Let us start all over again.0793

The best advice I have ever heard, when you lose your way, go back to the beginning.0798

Sec⁵ x, I'm going to write it as the integral of sec³ x sec² x dx.0802

There you go, now we are going to use integration by parts.0812

Now I go back to blue.0819

I’m going to call sec³ x my u, my sec² x is going to be my dv.0821

u is equal to sec³ x, dv is equal to sec² x.0827

du is equal to 3 sec² x sec x tan x.0836

v is just equal to tan x, very nice.0848

We know what the integration by parts formula is.0853

It is the integral of u dv is equal to uv - the integral of v du.0856

Here is my u and here is my v.0864

Now it is going to equal, this thing is equal u, u × v.0866

It is going to be sec³ x tan(x) - the integral of v du.0871

It is going to be the integral of tan x × 3 sec² x sec x tan x dx.0888

It is going to be sec³ x tan x - 3 × the integral of sec⁴ x tan² x dx, very nice.0907

We have this one already, we just need to deal with this integral right here.0938

Let us go ahead and do that.0945

I have got 3 × the integral of, we had sec⁴ x tan² x dx, that is equal to 3 × the integral of,0948

I have got an even power of sec, I’m going to pull out a sec².0969

I’m going to write that as sec² x, sec² x tan² x dx.0972

Sec² is equal to 1 + tan², this is actually equal to 3 × the integral of, let me rewrite this.0985

The sec², I’m going to write as 1 + tan² x × tan² x × sec² x dx.1000

u substitution, I’m going to let u equal tan x.1016

I’m going to let du equal sec² x dx.1020

Therefore, that gives me 3 × the integral of 1 + u², 1 + tan², u is tan.1028

1 + u² × u² × du = 3 × the integral of u² + u⁴ du = 3 × u³/ 3 + u⁵/ 5 + c1035

which = 3 ×, u is tan(x), tan³ (x)/ 3 + tan⁵ x/ 5 + c.1063

Our final answer, we pull the values that we had.1093

The integral of sec⁵ x dx is equal to, we had a sec³ x tan x, that was - this thing.1096

-3 tan³ x/ 3 - 3 tan⁵ x/ 5 + c.1115

There you go, that is it, just manipulation, that is all we are doing.1134

Let us discuss our last set of problems that we might see under the integral sign.1143

How to deal with integrals of this type?1148

When you have the integral of sin(ax) cos(bx).1151

The integral of sin ax sin bx, or the integral of cos ax cos bx.1156

You can use the following identities.1165

Sin(ax) cos bx, sin and cos where a and b are different.1169

Is the same as ½ sin of a - bx + sin a + bx.1174

Sin(ax) sin(bx) is ½ cos a - bx - cos a + bx.1181

Cos and cos is ½ cos, a - + cos a + b.1188

That is it, just using these identities.1194

Let us finish off with an example.1197

Evaluate the integral cos 5x sin 9x.1199

Let us rewrite it to make it look like it was, with sin first, it does not really matter.1208

It is sin(9x) × the cos(5x) dx.1216

A is equal to 9 and b is equal to 5.1231

The identity we are going to be using is sin(ax) cos(bx) = ½ × sin of a - b × x + sin of a + bx.1237

This equals the integral, this thing = this thing.1263

½ the sin of a – b, 9 – 5, 4x + sin of 9 + 5 is 14x dx = ½ the integral of sin 4x dx + ½ × the integral of sin 14x dx.1271

This equals 1/8 - 1/8 the cos of 4x – 1/28.1303

½, 1/14, 14 come out.1317

Cos(14x) + c, that is it, just using the identities.1321

Thank you so much for joining us here at www.educator.com.1329

We will see you next time, bye.1331