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Evaluate the integral below using any appropriate algebraic method or trigonometric identity I / 4 1 +4 sin 0cos...

Question

Evaluate the integral below using any appropriate algebraic method or trigonometric identity I / 4 1 +4 sin 0cos

Evaluate the integral below using any appropriate algebraic method or trigonometric identity I / 4 1 +4 sin 0 cos



Answers

Evaluate the integrals using appropriate substitutions. $$ \int \frac{\cos 4 \theta}{(1+2 \sin 4 \theta)^{4}} d \theta $$

This problem is from Chapter seven section to problem number forty six in the book Calculus Early Transcendence. ALS eighth edition by James Door Here we have a definite integral of the square root of one minus cosign for theta. So to deal with the square root, let's first deal with this co signing for theta. So coming over here to the right, we can apply the double angle formula and one way to state this is that co sign of two X is one minus two times science where so since we have we don't have a two x web of fourth Ada. We should use X equals to data. So then we have co signs for data is one minus two times science where the two data, then we have one minus cosign for data is simply to sign square to data. And this means that this way room will become square root too. Times square, the science where Susanna So one must be careful here because the square root of something square isn't necessarily the thing that's being swear. It's absolute value. So for that reason to be safe here, we'LL write one too Absolute value signed to theatre. So here, let's make an observation. Due to the bounds of integration, we have that data is between zero before. So this means that tooth data is between zero in power too. So multiply each side of this inequality by two and observed that sign is positive on the interval and separate us Ziona pirates who sinus positive there and sent sinus positive. There's no need for the absolute value sign so we could simply right I went to times scientific data, Islamics the thing that we're integrating much easier to work with. So coming back to the original problem we have in general sealed up our floor. I grew into stone scientific data separate our work on the side Also here in this new inaugural and my help you to go ahead and do it yourself here you could take you to be to theatre, not necessary. But it might help. So here we have negative won't too Coastline of two data. It's over too. And we have the end points. See our own power before, so it's gotta unplug those in. So it's plot this constant first, So plugging in pi over four, we have co sign a power too minus cause I know. Zero. So we have got you noting from the unit circle. This is zero. That's the one so determined. The parentheses is minus one and we have another minus one out here. So what you got? And cancel out those negatives. Take it square, too over to, and that's our answer.

In the problem we have in tradition. Got ta sign, Tita, did you? No cause day. But Jake, therefore minus sign. Tita, did you date was using here for minor division? Injected apart? Whole visit was when I was a kid. Born. Fine, fine. Blessed is even as minus one were five into God's. Oh, I'm kinda sissy System answer.

Were you called up a weather intercourse under Uba and EU. They wish you get encoded in uber and respond even in my embassy one let's see and in the question were given the integral on the coast I far Tita times square it under what do menace i for Tita Dita. And now it would just This one echoed you. So you cogitate humanise I for Tita, Then you ico Jude uh minus co sign Fontina on by the General when it attempts far in the front. So you had two terms farm here then Dita. And here it means that the DT time Here we go to the EU off minus far times co sign on the front it. And if I we should be able to write this integral in terms under you now. So cause I found Tita, stay the same. Square it on the new and detail because you did the year on Manus. Far cause I far, Tita. And then we say we can cancel the course I 40 time with this one. This one is a constant camping outside. Then we get echo to the minus one over far into co this Cuero we can register into the power half, do you? And now by formula, we get echo to monitor over far we and one to the power. We're gonna treatment of two on the tree and to see, But do you echo to this one? So we're looking, and then we get ICO Judah minus one out of six, and then we have ju minus sigh for Did, uh, about three on top, you see?

This question asked us to evaluate the given Integral. We know we can do this by distributing the numerator and simple farm before we take the integral. We just want to simplify the function like this. Now we know we can integrate because we know the integral of seeking scored of data is Tanith data on. Then one becomes data or variable, so pull again. 10 of pi over four plus tip over four minus 10 of zero plus zero gives us one plus pi divided by four


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