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Use the formulasin( 4au) +C 32asin? (au) cos? (au) du = 8to evaluate(sin? (3c) sin4 (3w)) dw.Include +C in your answer_...

Question

Use the formulasin( 4au) +C 32asin? (au) cos? (au) du = 8to evaluate(sin? (3c) sin4 (3w)) dw.Include +C in your answer_

Use the formula sin( 4au) +C 32a sin? (au) cos? (au) du = 8 to evaluate (sin? (3c) sin4 (3w)) dw. Include +C in your answer_



Answers

Use integration by parts to derive the given formula. $$\int \sin x \sin 3 x d x=$$ $$-\frac{3}{8} \sin x \cos 3 x+\frac{1}{8} \cos x \sin 3 x+C$$

Not having to find out this little being a one plus sign. So for the essay we have so just to find out that so one plus sign excellent for just work on this one plus sign excellent for that is humanized science can accelerate plus foursquare X rayed. Love to sign accelerate. Both exploring it's gonna sign accelerate plus both accelerate also the PM here now so now finding the root here and swear what we are integral wrote kind Extra very plus four actually old here, here's what's coming up to be, we'll get here now a lot of time accelerate plus because accelerate yes, so now what is all this? So we get here so we could be negative course next story or we have one over it and we have positive time. Excellent it or we have one or yes, let's forget now so it's coming up to be we have eight times sign exaggerated negative force. Thanks Sorry plus the this we're working on and this is gonna option the tenants or anything for a story adopting tv half we go with option C. Thank you

High in the question of Stephen F find an integral four calls especially Sine X. Over square root eight and design two X. Dx. And that is We compare the test right inside and find the value of Asia So how we can sold us so we can right here eight negative sign two X. As 9 -1 Plus. Scientifics that's coming out to be Integral. Call six negative cynics Dx over now route nine negative one plus in two weeks will be cynics plus cortex hold square We get to nine negative cynics plus core six old script. We can now use substitution here so we can use T equal sin X plus core sex. We got DDS course extended its sign next. So we got the integral and Bt over route nine negative. This girl. Now we know that the integral one hour route is Square negative excuse given a sign rules X over A. So that's coming out to be sign wills We have T over three so you're three forties cynics Plus co sex over three. The compare now we have here, Sinus was 46 times A over nine, So we have eight or 9 here, So it is 1/3 organized, 3/9. Compare this with 09 so we have a equals tweet. Thank you.

This problem is from Chapter seven section to Problem number five in the book Capitalists Early Transcendental Sze eighth Edition by James Story Here we have ah, indefinite Integral science of the fifth power of two t co sign squared of two tea. So here, since the power of sign is odd, the first thing we do is rewrite this by pulling out a factor of signs to have signed in the fourth Coastline Square. And let's put that extra factor of sign over here at the end. Chris, observe here in the inner grand that the science of the fourth power. So it's going to decide and rewrite this so we can rewrite sign square science of the four power as Science Square Square. Then we could apply, uh, put that in identity to rewrite sine squared is one minus coastlines where, and we have to square this entire expression and finally weaken. Evaluate this latest expression two co sign squared to team plus coastlines in the fourth power two teams, so replacing science of the forth. With this newest expression, we have the integral one. Minus two co signed square plus coast under the fourth Power Time's Cose identity to co sign squared to tease and sign off duty. So here it looks like we can apply u substitution here we should take you two be co sign opportunity, which implies to you is negative, too. Sign two t d t. That's using the chain rule and here because we don't have a negative to in front of the sign and the original under grand, we should make up for this, but by multiplying by a negative one half in front of the to you to give us exactly what we want. Sign two tea Titi Just as appears an immigrant. So now let's rewrite this integral using our U substitution. So first, let's pull off this negative on half outside the integral level one minus to use Claire. Plus, you know, the fourth then co sign squared is just use Claire and then to you. So now let's just leave this negative one half outside for now. And what's the stripy tissue squared inside the apprentices. So we have a use clear minus to you to the fourth power. Plus, you know the city's power to you to evaluate the center girl. We should supply the power rule three times. So we have you to the third power over three minus to you to the fifth over five. Plus, you're the seven over seven, plus our constancy of integration. So here we have two steps left will replace you with co sign of two tea, and we could multiply by this negative one have so we get a negative cool sign. Cube's Tootie over six. Here we have a plus coastline to the fifth hour over five. Those two's cancel. Now we have a minus co sign to the seventh hour, all divided by fourteen and closer constancy, and that's our answer.

This problem. We're gonna find the anti derivative of negative eight coast and of X minus seven. Sign of X d X. And because of the subtraction here we can do each of these parts separately. So the anti derivative of negative eight co sign X would be negative. Eight. Sign X and then the derivative of negative seven. Sign of X would be positive. Seven. Co sign X and then we add our constant of integration. Remember, we can always check whether or not we have the correct answer by taking derivative. So the derivative of negative eight sine x will be negative. Eight. Co sign x The truth of the seven Costa Next will be negative seven sine x And, of course, the derivative of a constant would be zero. And as we compare that to our original, we have the correct answer. So her final answer is negative. Eight. Sign of X plus seven coast and X plus C


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