for-each-pair-of-integrals-determine-which-one-is-more-difficult-to-evaluate-explain-your-reasonin-2

Question

For each pair of integrals, determine which one is more difficult to evaluate. Explain your reasoning
$$
\int \sin ^{456} x \cos x d x 	ext { or } \int \sin ^{2} x \cos ^{2} x d x
$$

Amy Jiang · Accepted Answer

Okay, so we have the inner Rose sign the power of 456 ex co sign of next e X and we also have to undergo sine squared X Coast critics the ex were asked and chairman, which one is more difficult to evaluate and explain that reasoning? Well, see, or the 1st 1 we can just pull out one of our actually, we could just use you was able to co sign of I Love go sign Sign of X The X then you is equal to Or do you is equal to co sign of x The X So we have what we want That&#x27;s it. Ut 0456 in a row to you that&#x27;s That&#x27;s you to the 457 But you a sign of X to power 457 over 457 seek. Okay, that one&#x27;s pretty simple. Let&#x27;s see about the integral of sine squared X Co sign for ext the ex con Cut. We can use our assigned toward Accent Coast and scored ex identities. That&#x27;s the people to one minus co sign two x over two and one plus co sign of two ext over to the X and then I can factor out a two are these two. So that&#x27;s one over four. And then I take the products of these two. Or what I could do instead is that one mine schools into X times one plus goes into accidents one minus co sign squared to X X. Well, based on this information, we can already tell that the seconds what this integral is gonna be a lot more work, kids. But this one, we only needed, like, two steps. But this one already we have three, but we still need more steps to solve it. Okay, listen. To solve this, we have our property for co sign Squared X, which I was in court to x interview of one minus 1/2 but it&#x27;s one over to sign up for X X. Okay, so that&#x27;s one over four X money, Mrs 1/2 X over two minus one over eight. Sign of four x plus C as a solution. Yes. So the 1st 1 is a lot easier for the 2nd 1 We needed to use our properties or our identities to translate it. Our science cortex and clothes and Codex. And then we had had to translate it again over here, and it became this and it would take the indigo

Yaw Asomani · Answer

suppose you have a tangent to the power. Anything? 1000 Billy. Anything. Okay, its longest you have. The derivative of attainment attached to it is very simple. OK, so over here is so simple because the derivative off tangent, which is seeking squared, is attached. So you just have to let you be detainment. That&#x27;s it. And the derivative of the tangent is seeking squared xcx. Okay, so you see that the derivative of the tangent is right here, so you can easily make the substitution. Okay, This one is the same ass. Just standing here is the same as you. Okay, Let you be the tank. It&#x27;s so in place of tank, and we&#x27;re gonna put you. But you has to be having 3 50 as the power because it standing to depart with 3 50 So it is gonna be to 50. And then this guy right here is Do you from here? Okay, saying place off that guy. I&#x27;m just gonna put you and this becomes a simple integration. A u to the power 3 51/3 51 you know? So So it&#x27;s armed it okay And all that was made possible because the derivative oft agent is seeking squared. OK, but if you have something like change into 3 50 X and then just seek and okay, you can see that the derivative of the tangent is not seeking it is seeking squared. OK, also, the derivative will seek and it&#x27;s not change it. It is Egon exchange in X so you can see that doing a substitution here is gonna be extremely difficult. You can possibly due a substitution. You have to probably do a different thing or you have to, I mean, flu for different combinations of techniques. But this one is pretty straightforward about three or four lines you&#x27;ve done all because this one is the derivative off tangent affects. Okay. And this one is difficult because this one, it&#x27;s not the derivative off this. OK, so this one is gonna be easier to evaluate, and this one is not being a lot harder to violate. Okay,

Amy Jiang · Answer

Okay, so we want to take the interval. The following we see how we have sine squared rex. So we want to reduce our power. So we can do is we can use our half android, That is We know that, um, sine squared of x is able to mourn at minus co sign two x over to. So let&#x27;s rewrite that in terms of coast on two X. So we get the integral weaken Split up this into two fractions. That&#x27;s 1/2 the X minus two in a row of CO sign of to over to the X. So that gives us, um, X over two minus, um co sign the owner of Crow Sinus Sign of two X over two times two, which is four. And then we have plus c. Okay, so we get the full insulation.

Amy Jiang · Answer

Okay, So? Well, it is all for four in a row. Notice that we have two powers of two, and either Even so, we can&#x27;t really switch them off on and use. Use that instead. Let&#x27;s use our half. Requires anything. Well, for sign. That&#x27;s what mine is co sign to over two on for, um, coastline. That&#x27;s one plus close to of the two. That a knowledgeable sweat it out. We get in the girl. Uh, one minus co. Sign. Word of two. Number four. I will bring out our board. That&#x27;s 1/4. Okay. And I was used or half ago again. One more time. Who wouldn&#x27;t want four in a row of one minus? Uh, this is not one one plus co sign of two times two accidents for over to. So we have one minus one house. That this 1/2 And then plus a closed sign for two. The X. Okay, I&#x27;m not taking her in a row. You get 1/4 times, um, one of its you close. Then you were closing that sign before over four times to That&#x27;s and we have plus three. This is our fallen solution.

Audrey Fong · Answer

we are to determine which of these two into grow can be integrated using the max as well. And so far it would be this integral. Of course we&#x27;ll sign square. Thanks. It can be integrated with other techniques, but not with the metas. Well done. So far so far. Signed to eggs. Were you substitution? Let U equals two X. Do you will be through the X. So I&#x27;m missing it. Do here. What I can do is I can put a hot I&#x27;m signed. Two weeks. I can put my two d x here for now. I&#x27;m ready to substitute two extra you into DS Me? You. So I Huh? Signed you. Do you sign? When you integrate, you&#x27;ll get my new school sign. New plus c replaced. But the you will two x So I minus half course sign two X rusty

Amit Srivastava · Answer

integration. I equal toe the sign ICS And if you take woman as you call sex, forget it. Go six Bless one air DX Now let cause sex equal to he. Hence the physician off. This will be minus sign. It&#x27;s three x you could do dating with the final part will be minus dirty by they indoor replacement now using pasta perfection It is a by a plus be by people So we d plus one now b d And this is your minus one. If we take equal do zio, then it will be minus one. If Deke will do minus one, then be quarto. So finally digression will be a appointee Men&#x27;s and minus one one Body day T and the bees one. Do you buy replacement from minus log P Let&#x27;s look people s one Plessy empty will be replaced by Cossacks. So finally finally it will be logged called sex bless log courses plus one plus

Problem

For each pair of integrals, determine which one i…

Problem 124 Medium Difficulty

For each pair of integrals, determine which one is more difficult to evaluate. Explain your reasoning
$$
\int \sin ^{456} x \cos x d x \text { or } \int \sin ^{2} x \cos ^{2} x d x
$$

Answer

(1) $\frac{1}{457} \sin ^{457} x+c$
(2) $\frac{1}{4}\left(\frac{x}{2}-\frac{1}{8} \sin 4 x\right)+c$

Related Courses

Calculus

Related Topics

Discussion

Top Calculus 2 / BC Educators

Catherine R.

Caleb E.

Kristen K.

Joseph L.

Calculus 2 / BC Courses

Integration Review - Intro

Definite vs Indefinite

Recommended Videos

Watch More Solved Questions in Chapter 3

Video Transcript

Gilbert Strang,

Calculus

Catherine R.

Caleb E.

Kristen K.

Joseph L.

Integration Review - Intro

Definite vs Indefinite

What do you need help with?

Textbooks

Ask a question

Courses

AI Tutor

(1) $\frac{1}{457} \sin ^{457} x+c$(2) $\frac{1}{4}\left(\frac{x}{2}-\frac{1}{8} \sin 4 x\right)+c$

Calculus

Catherine R.

Caleb E.

Kristen K.

Joseph L.

Integration Review - Intro

Definite vs Indefinite

Gilbert Strang,Calculus

Catherine R.

Caleb E.

Kristen K.

Joseph L.

(1) $\frac{1}{457} \sin ^{457} x+c$
(2) $\frac{1}{4}\left(\frac{x}{2}-\frac{1}{8} \sin 4 x\right)+c$

Gilbert Strang,

Calculus