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Question

Solve the given problems by integration.
Integrate: $\int \frac{\cos x d x}{\sin x+\sin ^{3} x} \cdot($ Hint: Begin by making the substitution $u=\sin x$.)

Vikash Ranjan · Accepted Answer

in the problem we have been given aggression call 60 X. Upon cynics plus thank you X. So U. Is equal to cynics do you becomes Kazakh sticks? Therefore it is to listen do you upon you plus you have the power to you? Therefore it is equal to depression. Do you upon you into U. S. Squared plus one? No filter. We have to solve this so this can be further represented as depression. Do you upon U. To the power Q. Is taken out so it is one class went upon us squad now put one plus one upon us choir as in jail. So it becomes You&#x27;re the par -2 that equals one plus you two for minus 2 30 goes dead. So you know -2 You depart -3 Do you? That equals two digit. Hence We have this equals two minus two upon U. To the power three. Do equal dessert. Or this becomes mine is going to point to a direction digit upon jade. This equals minus Went up onto Ellen martin jade. So we have this equals two. Mine is going to point to Ellen duties once upon sorry this is one plus one upon us. Quiet plus C. Not for that. This is written as a -1 upon to Ellen. You esquire Plus one upon he was square. Classy. I found that it was too management upon to Ellen You esquire plus one minus or it is taken out so that&#x27;s become minus Ellen U. To the power two further disciples minus went up onto Ellen mud U squared plus one plus half. Ellen the powerball plus C dizzy place -1 upon to Ellen mode us square plus one plus Ellen you plus the from that this has given us Ellen model cynics -1.2. Ellen mud sign square X plus one plus C. So we have this as the answer to the problem.

Vikash Ranjan · Answer

the problem. We have to find indication off. Signed to the power tricks. Koszics. Here he is u U equals sign it Therefore, this deal will Kosik sticks hence it is a return as you powertrain. Do you therefore these you the powerful or full? Let&#x27;s see, we can write it as a sign to the power forex. Well, one full blast. See, it is the result to be given a problem.

Darshan Maheshwari · Answer

we need to value the fall following integral using the given substitution. So if you assign X if you differentiate both sides, we have do us Cossacks DX so let&#x27;s read that are integral sign is replaced by you. Were you Cube Cossacks TXs to you So this integral becomes your ears to the poor four or four plus. See where the value of us cynics, We have signed Reyes to the poor forex. Okay, over four plus c. This is nonsense.

Amy Jiang · Answer

Okay, so we want to find the falling integral by using use up and were given or fall on you, William. So let&#x27;s start by finding, do you? But that&#x27;s equal to the derivative of sign, which is co sign of X. And then we also have DX and ours here inner integral. We have coastline of X d. X so we can rewrite this in terms of you that is the integral sign of X that&#x27;s you. And in Costa X, the X that see you So that&#x27;s just equal to you Squared over two and m plus c analogy face, are you? We said you is equal to sign a back. So that is we get sine squared of X over two and M plus C for falling solution.

Karuna Ramroop · Answer

All right. So this integral, you&#x27;re gonna want to be familiar with your trick functions. So the first thing I like to dio when I see trick functions is I like to draw this little circle of derivatives between finding co sign because it&#x27;s helpful. Um, I know that they&#x27;re derivative of sign of ex aides. Coastline of X. I know that the derivative of co sign of X is negative song avec nous. I know that the derivative of negative sign of X is negative co sign of X and I know that the derivative of negative co sign of X is side effects. So this is a circle of derivatives. But the nice thing is, it&#x27;s you could also think of it. It&#x27;s a circle of inter girls if you just go in the opposite direction going clockwise were shaking into girls. But going counter for wise we&#x27;re taking, I&#x27;ll introduce Sorry, going clockwise. We&#x27;re doing derivatives going counterpart boys were doing into girls. So first thing we&#x27;re gonna do is figure out what the intervals of sign and negative coastline are. So for sign, we can see that the intra bro just going counterclockwise. That gives us negative coastline vex. So this is going to become negative co Sign of X. Since there&#x27;s no term upfront, since there&#x27;s nothing inside of the parentheses with the ex, we know that this is just simply negative coastline of X. And then the next thing you wanna work at A&#x27;s are negative posts and events. So the integral of negative co sign a lex is negative side effects. So this is gonna become a minus sign of X, and this is evaluated from zero. So the next thing you wanted you is plug pie into this expression. So we&#x27;re going to get I&#x27;m gonna put this in some big parentheses to make it a little bit easier to read. Negative co sign Hi. Minus sign of pie. And then I&#x27;m going to choose a tract. Another big parentheses. But in this one, I&#x27;m gonna put zero into the expression. So this is gonna become negative. Who signed Jiro minus sign zero. And I&#x27;m just using the rules of integration with the definite in general like this. Once you get to this stuff, all we&#x27;re doing is we&#x27;re plugging o R are taught number into the expression minus our bottom number in the expression so ho sign of pie is gonna be negative Born the time to use a calculator or you could have virtually no political So we have a co signer pies Negative one. This becomes negative. Negative One sign of pie is zero. So this just minus zero and then or our second front sees we have negative co sign of zero Well, co sign of zero is just one just becomes negative one minus Whatever side is you always and we know that sign of zero is stressed here So then stop buying this further. Uh, we have negative negative one. Let&#x27;s just becomes a positive one. Positive one minus zero. It&#x27;s just positive one. So this whole first parentheses just becomes what than minus. We have just negative one in the second prophecies because negative one minus Syria was just so this becomes one minus negative one. Well, two negatives make a positive. So that becomes one plus one just and that&#x27;s your answer

Pagadala Kishore Reddy · Answer

look it. So we&#x27;re going toe. We You&#x27;re really hurting, though. And do the laugh. Thank you, Bix Koszics DX. We know that the baby well say next equals Koszics. So we evaluated this integral bye U substitution matter. So the next thing is lead you equals say nix No d&#x27;you equals that every radio outside next is car six and that very creative off excuse the ex. So degree love saying que Bix car six d x equals sign. Is you so saying cube is you Q and Koszics VX. He&#x27;s do you and we know that in do the love export in B X equals export and plus one by n plus one plus c so integral a few q b u equals you forward here in the street So three plus one by three plus one Let&#x27;s see that equals you. Powerful way cool Let&#x27;s see requires we know that you is say nix Saint ex playing board for ticks before bless seat so integral off sine que bix koszics DX equals same power for ticks before. Let&#x27;s see. Thank you

Problem

Solve the given problems by integration. Find the…

Problem 26 Easy Difficulty

Solve the given problems by integration.
Integrate: $\int \frac{\cos x d x}{\sin x+\sin ^{3} x} \cdot($ Hint: Begin by making the substitution $u=\sin x$.)

Answer

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Allyn J. Washington, Richard S. EvansBasic Technical Mathematics with Calculus

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Allyn J. Washington, Richard S. Evans

Basic Technical Mathematics with Calculus