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Solve the given problems by integration.Using the identity $\sin \alpha \cos \beta=\frac{1}{2}[\sin (\alpha+\beta)+\sin (\alpha-\beta)]$integrate $\int \sin 4 x \cos 5 x d x$

$\frac{1}{2} \cos x-\frac{1}{18} \cos 9 x+C$

Calculus 1 / AB

Chapter 28

Methods of Integration

Section 5

Other Trigonometric Forms

Integrals

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to let us have signed five to the X. Okay? Times co sign. I swear. X X. Okay. Uh, we know from trigonometry that sine squared X plus co sine squared X equals one gate. So co sign here. Script Coastline squared X here can be written as one minus sine squared X. Okay, so we're gonna change this, consigning to sign. So this is the same as saying five x one minus sine squared. Execute the X from this identity. Okay. And they in this is gonna be sign. I'm just gonna foil. Okay. Sine x minus. Sign to seven. Okay, we have this one. When you foil this one like that, that is what you have. Okay, So once you have that, what is happening, then? This is gonna be sign five x dx minus signed to the seven X also de XLs. Consider them one by one. Okay, let's consider this one. So when you consider dad one, this one is gonna be, uh, you know, signed to the four X eggs, though. Scientific four x times sign X k sine x dx. Okay. This scientific I'm asking, broken down, as is this and then signs in a four X is Emma's signs where X to the power two. Okay, sent X dx. Now I'm gonna break this 12 into co sign using this identity again. So he's gonna be one minus school sine squared. X swear sine x e x. Okay, then this one is gonna be I'm gonna do square. That was where I'm gonna expand this. Parentheses is gonna be one minus twice. Hussein squared x of course, co scientist of fourth Power X Kate and then sign X DX. Okay, So once you do this, then you're gonna let you right equal to co sign X. Okay, then what is gonna be determinative? Do you is sine x? Yes. OK. Therefore, the X is gonna be do you over a sine X when I divide those side by signing exact Get this one, Kate. So from here, I get to this one. It was already see, Costa and X could use you. So this is cool San squares. There's gonna be used weird. And then this is consented of force is gonna be you 2 to 4. Hey! And then this sign eggs is gonna be here. And then this the x here is the same as to you over a sine X right? So I'm gonna put their do you over sine X. So this one cancels this one. Okay, then what I have is one minus two squared. Plus, you did a four k the you, which is a simple integration. So this one is gonna be U minus two or three. You did it. Power three. And then ah u to the power 5/5. OK, but what is you? You is co sign X. So this one's gonna be co sign X minus 2/3 co sign Cube X plus co Scientist five, Expo five. So that is just for this portion. That is just for, uh, this portion k. You're gonna apply the same trick. You see, the problem is that when the power is an odd power, you can apply the same trick for it. Okay, so we're gonna play the same trick for this one as well. Okay, so now we consider signed to the senate. Get it? Power is odd, so it's easier. So that is the same as sign to the power six k time side X dx already know this one. Okay, so, uh, you're gonna convert this one? Okay? This one is the same as, um, this one is the same as, uh, sine squared. X cubic. Okay, I'm rewriting. I've written this one as that is the same thing. Okay. Okay. So and I'm gonna converted to co sides. That is gonna be one minus co sign square extra power, three gay side xcx. Then I'm a let you be co signed X. Okay, then do you is gonna be negative. Negative sine x. Okay, so when I let you be negative to use negative sign eggs, DX. Okay, so D eggs is gonna be negative, Do you? Over a sine X. So when I bring it here is gonna be one minus use were OK, because I let you become saying X circle science. Where is gonna be used where this is three. And then this one is I'm gonna bring the negative out, and this one's gonna be to you because this sign has been a cancel. The sign. Right. So I'm gonna have dissed eu right here. Okay, So once you have that, then we're gonna use Moscow triangle to, uh, expand this one case it was triangle right here. Okay. Is this one okay? So is gonna be one. Mine is Ah, three u squared plus three u 2 to 4 miners. You to the sixth. Okay, That is a bus, Kyle for you. And in the EU. Okay, we doing it to you. So here is Do you? So then the integration finally is gonna be, uh, u minus you cute plus 3/5. You to file. Okay. Minus you 2 to 7/7. Okay. And then this one is gonna be, uh, you know, it is negative here is gonna foil through. Okay, so it is gonna be negative co sign X, because I let you because I explicate. And then this one is gonna be positive. Co saying cube X minus 3/5. Close sounded at five x then plus co sign to the seven. Expo 70. That is the integration off this guy right here. Okay. Signed to the 7/7. I made a mistake here. Here. Do you is supposed to be negative. Sign again. Okay. So the negative is gonna prevail here. Okay, so, uh, that negative is gonna make here negative, Positive and the negative. Okay, Now, since I'm done with two integration now have to combine the like this. Like so Okay, this is the original thing. So I have to combine them with this original thing right here. Okay, so the combination is gonna be, uh, this OK, minus that, right? I don't think I have enough board to write that. But, Aiken, see it straight away. OK, so when you take this minus this cancer, is it right? Cause he's gonna be minus minus. Is is gonna be positive. OK, so, uh, let me just rename it like this so that you can see exactly what is going on. This is equation one. Okay. And this is equation, too. Okay? And we're doing, uh, equation equation to minus equation one. Okay, so, uh, but the equation to minus equation one, then negative co sign X minus minus CO sign eggs, zero k. So to go away, they in, uh, Tshosane Cube, X OK, minus 2/3 coast. Thank you. X. What do you have? You have 1/3. 1/3. Co signing Cube X. Okay. And then Ah, this negative three over five. Close on five eggs. minus minus. Okay. Co signed five X over five. What is it gonna be? So negative? 3/5 K plus 1/5. It's actually what we're doing. That's gonna be negative for over five. Okay, Negative for over five. Co sign five X, then. Plus. Now this one has no counterpart here. Okay, so that is just gonna be written right there. Okay, so this one is just gonna be won over seven. Okay, Coincide. Seven X then plus C. That is our sign five x minus. Sign seven x dx. And and mind you, this one is the same as our integral sign five x co sine squared eggs, The exit gates of the integral off this thing right here is all of that. Okay, so that is the answer to your question. Okay? This is the answer. Very long a

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