Integration-Pt 1-Transforming an Inverse Trigonometric Expression into an Algebraic one.

THE CONDITION FOR UNCONDITIONAL:

1. All my math hubs have something about God in their final paragraph(s). Paragraph 12 of this hub addresses the condition for God's unconditional love.

INTEGRATION AND MATHEMATICAL MANIPULATIONS:

2. There are two main branches in calculus: differentiation, and integration. To do differentiation in a 1st year calculus course requires following the rules and doing the arithmetic. To do integration you have to be a sage, a prognosticator, a clairvoyant, and mostly a by-guess-by-golly type of person. You take a guess, and by golly it worked. Some of the techniques to do integration are absolutely brilliant. Integration by parts, logarithmic integration, Taylor expansion, converting to polar coordinates and trigonometric substitution are just a few of the many techniques developed, and which can be used to perform integration. Although I may be easily impressed, I think they are all very clever. They all boil down to being mathematical manipulations which can make evaluation of the integral easier.

WHAT IS TRIGONOMETRY IN A NUTSHELL?


3. Well, it depends on the size of the nutshell in order to easily explain it. Let's make it in the shape of a rectangle with 625 feet for its length, 104 feet for its width, and the roof will be an all glass cover in the form of an ark--oh excuse me-- arc. The length will run south to north, and if you are lying in a sleeping bag on your right side at the extreme southern end, you want to be able to open your eyes and see the north star. You happen to know that the angle you must turn your eyes from the floor is about 31 degrees( 31.2333 to be exact) in order to see the north star. So what height must you limit the walls enclosing this rectangular "nutshell" so that you can see the north star from the southern end? This is what trigonometry( trig) does. It relates the 2 sides( the floor and the wall), and also the hypotenuse( the diagonal line directly from your eyes to the tip of the northern wall) to one another. Having knowledge of how those relationships work together is what trig is all about, and it enables one to solve an enormous amount of problems in physics, engineering and many other fields. In this case the wall must be 625( tan 31°)=375.54 feet, or round down to 375 feet. I suppose I could have made this nutshell 120 times smaller and still made my point, and even at that size I would still have plenty of room to put all my important papers. Well, I have to plant a tree 34 miles east of here, but I only have 47 minutes to get there. Doesn't sound too tough but all I have is my horse, and even Secretariat could not run that fast.

INVERSE TRIGONOMETRIC FUNCTIONS:

4. Entire textbooks are written about trig, and several chapters are dedicated to the subject in precalculus textbooks; therefore, in this hub we will only take a peek at some of the pertinent concepts.

5. At L1 cot stands for cotangent, and cot--1 stands for inverse cotangent, or arc cotangent. cot--1 represents the angle for that value of cotangent, and it is another way to write the angle, θ. One way to think of it is "arc" is the arc made by that angle, θ( or cot--1). So the notation at L1 means the same thing as the notation at the first line of L7. The second line of L7 is a trigonometric identity( trig ID) that equals the first line.

FIGURE 1 EXPLAINED:

6. Transforming an inverse trig function to an algebraic one is a very effective way get it in a form that you can integrate. Trig is fundamentally all about the right triangle. When you can get this in your head, then you are well on your way to do the manipulations necessary to do these conversions.

7. Figure 1 represents (1/4)th of the unit circle in the 1st quadrant. The unit circle has a radius of one; therefore, the length of line AC is 1 unit( foot, mile, inch, cm, meter, whatever). Both cosine( cos) and sine( sin) are defined as AB / AC, and BC / AC respectively, but since AC is 1 then they reduce to what is shown in figure1. The angle, θ, is, in this case, about 31 degrees. The terms at the bottom of fig.1 are those normally used in trig to describe the 3 sides of a right triangle. Arc CD is the arc on this unit circle that is made with angle θ.

L1 THROUGH L5 EXPLAINED:

8. The first thing to do to convert L1 to its algebraic equivalent is to define the angles at L2 and L3. You may have read that a function is like a machine that operates on something. You put a value into it, and it cranks out another value( sometimes the same one). This is what the cotangent( cot) function does. You put a value for the angle, θ, and it cranks out a value for line AB divided by BC. Trig has been developed to the point that we know what the lengths of AB, and BC will be at a specific angle. At L2 if we take the cotangent of the left equation then we will get the right equation. Taking cot( cot--1u) cancels out leaving just u. It is similar to multiplying 13 by 1/13 and you are left with 1. Now we know that u is the value of cot θ1. That is telling us that AB / BC is u. The only way it can be u is if line BC has length 1, because any number divided by 1 is that number. So we put that length on L4, and we put the length of u onto the adjacent leg at L4. We then calculate the length of line AC using the Pythagorean theorem, and place that at L4. We do all of that for θ2 at L3. I have a subscript of 1 there at the left side but it is supposed to be 2. The math here is forcing us to leave the unit circle, right? We already have length of 1 for BC; therefore, in this case the length of AC must be greater than one.

L6 THROUGH L8 EXPLAINED:

9. L6 is just a reminder of how cotangent is defined. As I said at paragraph 5, L1 and L7 say the same thing; therefore, we write it in standard form at L7, and then give the trigonometric identity at the 2nd line of L7, which represents the 1st line of L7. The rest is really easy. We know the representations of sine and cosine given at figure 1; therefore, we plug these values of them at L4, and correlate them with the 2nd line of L7, and place them on the 3rd line of L7. L8 is just doing the arithmetic to simplify L7, and our final answer is at L8. This is the algebraic representation of L1, and it is in a form that we can integrate.

L9 THROUGH L15 EXPLAINED:

10. At L9 csc stands for cosecant, which is the reciprocal of sine; therefore, it is( from fig 1) AC / BC. So L9 says to square the value of csc θ. We already know how to represent arccotangent X. Here cot -1(X) is just being used to represent the angle, θ. The triangle at L10 was explained at paragraph 8. Sine θ is defined at L12. Since cosecant is the reciprocal of sine then the 2nd line of L12 represents that. After squaring that value we have the final answer at L15, and this is very easy to integrate. Integrating we have ∫ X2 + 1 dx gives and answer of (1/3)X3 + X + C( a constant). L9 would have been much more difficult to integrate if we had not converted it to its algebraic representation.

MUCH EASIER INTEGRATION IS ACHIEVED:

E1. These are edits I finished on 11-19-2012: You can try and integrate L9 without first transforming it into its algebraic equivalent, but it would be much more difficult. At EL1 I write L9 in a form that my Ti-89 will recognize. At EL2 I tell the ti-89 to integrate it, and I get EL3. The integral sign applies to the entire fraction, not just the denominator. Notice my ti-89 did not integrate it at EL3. If I put in limits of integration as at EL4, then the ti-89 can give me the answer of 2.0253 at EL4.

E2. Look how much easier the integration is at EL5 through EL7. The integrand at EL5 is the algebraic equivalent of L9, and it is so easy to integrate, one can do it without a calculator. We get the same answer at EL7.

E3. EL8 is showing that EL2 and EL3 are the same function with the angle in the first quadrant. The ti-89 gives, of course, the same answer.

E4. You must be aware of the domain when doing these transformations. I kept the angle in the first quadrant to keep it simple.

WHAT IS NEXT IN THIS SERIES?

11. The final answer at L8 will be integrated in part 2 of this series on integration. We will look at the difference of θ1 and θ2 being two different angles on a plane, or two different angles in three dimensional space. In future hubs I'll address some of the techniques of integration mentioned at paragraph 2.

WHEN IS GOD'S LOVE UNCONDITIONAL?

12. This is a continuation of paragraphs 12, 13, 14, 15, and 16 of hub#12.9. I believe God's love is unconditional concerning who can repent and then be saved. Scripture is very clear on this. God is not willing that ANY perish, but that ALL should come to repentance( 2 Peter 3:9). It does not matter who you are or what you did: all means all. God would not command ALL people to repent( Acts 17:30) if He did not expect that all people had the ability to repent; nor would God send the "all likewise perish"( Luke 13:3 and 5) people to eternal hell if they did not ALL have the ability and choice to live eternally in heaven with Christ if they repented. The offer to choose Christ is to all people( Joshua 24:15; Deuteronomy 30:15 to 19; 1Kings 18:21). Whosoever means whosoever; therefore, John made it very clear that ALL people have the ability to choose live, or death; to choose Christ, or sin; to choose heaven, or hell( John 3:16; 4:14; 11:26; 12:46; 1John 3:6 and 9; 4:15; 5:1 and 18; 2John 1:9). So in this respect God's love is unconditional---ANYONE CAN CHOOSE LIFE! There are no conditions given with these verses. None of them say all or whosoever, except those of you who did this or that, or did not do this or that. People come to Christ at different times in their lives, and you can be sure that many of them have had a questionable( to put it mildly) past. Regardless of how sinful one's life has been, God's love is unconditional--according to the Scriptures given--concerning God's offer for that person to choose life( i.e. obey Christ; obey God's Word), and be saved from eternal hell, and to eternal joy and rest in God's kingdom.

More by this Author


Comments 10 comments

Caleb DRC profile image

Caleb DRC 4 years ago Author

I'll bet you didn't think I would beat you at the punch on the comments section, did ya, North Wind?

Well, as you know, if I have a choice on the numbers I use in my hubs then I choose numbers with specific meanings. There is enough information in paragraph 3 for you to identify 2 very important structures in the Bible, and to give the exact location of the nutshell. Oh, perhaps I should limit the location to be on the surface of the earth. You know, I'll bet Abraham can help you with this.

This is not a challenge you need to accept, North Wind; after all there are only so many hours in a day. I've decided to do this on all pertinent hubs. I think it will be fun for some people, but more importantly it will get one to think. Anyone who seriously reads the Bible has to think a lot! Ain't that right?


North Wind profile image

North Wind 4 years ago from The World (for now)

I doubled over laughing when I read your comment, Caleb. I was literally thinking,'Who beat me to the comments?' HA! I thought paragraph three was a clue and was actually going to ask about your play on words (ark and then arc).

My mind is actually spinning, Caleb. I think I know what you are talking about but I do not know if you realized how much sense your numbers make. God's word is like this intricate spider's web that leads you to almost every thing. Just curious - did you know when you typed 625 that Genesis 6 speaks of Noah and Genesis 25 speaks of Abraham ? Also, somehow, I have visited Sinai and the pyramids through those numbers as well as Turkey.

The Ark of the Covenant as well as Noah's Ark both came up as I decided to widen my search as well as the Garden of Eden. I confess that I got very distracted by your description of the arc because I could not help think about the firmament called heaven that divided the waters.

I have to think some more and examine the numbers and words as well. Sometimes, numbers are in words too just as words can be in numbers.

Not a challenge I need to accept? Every time I read one of your hubs my brain goes involuntarily into super drive and I find myself reasoning out Math and connecting it to the fibers of the world. I still haven't finished thinking about Math and Science and their relationship to one another. I think Math is more than a tool to Science. I just can't get around it. I think that Math works for Science because Math is actually Science's child or a sibling maybe. I can't explain it except that I know that God can wrap everything up into a tiny speck and show how everything is all related with just one word. There is just so very much we don't know. Knowing that helps me to have great awe for Him and I am grateful for that.

Anyway, I will definitely be thinking about this and doing some Bible study on your numbers and words so that I can find that nutshell!


Caleb DRC profile image

Caleb DRC 4 years ago Author

Wow! North Wind. You got the two structures almost immediately: Noah's ark and the Ark of the Covenant. Our nutshell is the same size as Noah's ark except for the height. I could not make the nutshell only 62.5 feet high because it would have given me only 5.71 degrees for an angle, and that was too small for figure 1. The Ark of the Covenant has a length of exactly 120 time smaller than Noah's ark and a height of 120 times smaller than our nutshell. Bullinger can tell you what the 120 means. Dake's Bible says that Noah's ark could carry 90,000.000pounds( 600 freight cars making the train 4 miles long).

You have a creative mind---really! I did not think of half the stuff you did. I was not aware of Genesis 6( Noah), and 25( Abraham) and the 625 length. Nor did I think of the firmament and the glass roof of the nutshell. I just needed something to keep the rain out and still see the North Star. Come to think of it we have Job 37:18 associated with the glass, but I just thought of that. I think the firmament is much more complex than we think. It is more than just sky, and I think it has some very esoteric physics associated with it. It is like Job 38:19--AMAZING! This implies all light has been created and the nuclear processes in stars release the light, not create it. Quantum mechanics can be applied to the same reasoning for candles and flashlights. And "the way where light dwelleth?" A specific way the electromagnetic force accesses it? Analogous to Christ being the way, and the Light.

Yes, math the child of science--or the other way around. I believe God structured the mathematics first, and then made creation based on the math. Just in Isaiah 40:12 alone we have 5 references to measurement, and Oh! so much more throughout the Bible. From this perspective, math is science; science could not exist without it. Yes we had science before we started attaching numbers to it, but that does not mean the numbers did not exist prior to the science, just as we have had DNA in us long before we discovered it.

I read somewhere that in the Hebrew language the letters also have numerical meaning, and books have been written about the fascinating research done when the numerical meaning is applied throughout Scripture.

You're pretty sharp, North Wind, and I think you will have the location of the nutshell soon. Abraham and Abimelech will agree on that. I got that tree planted. I was so surprised I could go 34 miles in 47 minutes on a horse, I thanked "the Everlasting God."


North Wind profile image

North Wind 4 years ago from The World (for now)

Beersheba! The place where Abraham and Abimelech made a covenant and where Abraham made sacrifices to the Lord God. I knew there were coordinates in there and you would not believe what I found when I looked up those coordinates the first time. I see now that Beersheba is located at 31 degrees but I am very overwhelmed, quite frankly, by what I found at first.

Anyway, I still did not get to buy Bullinger's book but I did find that Beersheba is 120 kilometers south-west of Jerusalem. Could you tell me what he says about the number?

I can only conclude that you could manage that speed through the help of God as Elijah did when he outran the chariot of Ahab. I only now noticed the trees that Abraham planted. Elijah would have ( could have) rested under one of them when he fled from Jezebel ? I wonder if those trees are still there?

Do you know that when I opened my Bible to the Psalm I had been reading I found two references to the horse:

"An horse is a vain thing for safety: neither shall he deliver any by his great strength." Psalm 33:16 and

"He delighteth not in the strength of the horse: he taketh not pleasure in the legs of a man." Psalm 147: 9.

I marveled at that because there was no way you could know that I have been studying the Psalms.


Caleb DRC profile image

Caleb DRC 4 years ago Author

Exactly, North Wind. Beersheba is 31 degrees and 14 minutes north by 34 degrees and 47 minutes east. Abraham planted a grove( tree according to Strongs) there and called upon "the everlasting God."( Genesis 21:33). It was also there where Abraham and Abimelech made a covenant( Gen. 21:22--24). Many other things happened there.

What did you find "by what I found first"?

120 is three forties( 3 X 40( a significant number itself in Scripture)), and represents "a divinely appointed period of probation( Gen. 6:3)." Bullinger also mentioned it is a factor of many numbers in Scripture.

The horse is a magnificent animal( Job 39:19--25), and he glorifies God. I read Psalms 33:17 and 147:10 saying that since the horse is such a magnificent creature, then where does that put its Creator on the scale of magnificence? I could not use a car to make that distance because a car is too fast, but a horse was perfect. They are mentioned several times in Scripture, and I truly do think they are magnificent( Watch "Seabiscuit" with Jeff Bridges, and "Secretariat"). Watching those scenes where they are racing is just plain cool! In researching this I found that Quarter Horses can run that fast( 43.4 miles per hour or 34 miles in 47 minutes) but not for that distance

Good work, North Wind. I think in your case I'll have to make the riddles tougher. Not only are your hubs very good, but they are very beneficial to readers with eyes and ears; therefore, when you set out to solve one, just don't allow it to take time from your own writing because you are doing God, and society a good service with your hubs here.


North Wind profile image

North Wind 4 years ago from The World (for now)

Thanks for the encouragement, Caleb. Yes, the horse is a magnificent animal. I never saw Seabiscuit or Secretariat because I thought that the horse might die in the end but I'll put them on my must see list.

As for what I found first - it was just too funny and weird at the same time. I found something to do with me personally. What are the chances of that happening?

I loved doing this riddle and look forward to the next one but don't make the next one too hard OK? :)


Tomás en España 3 years ago

Is to me confusing in the last part, is antiderivative of csc^2(cot^-1(x)) = (3x + x^3)/3 + c? I follow the pictures, but text part is confusing. I don't know of calculus of Jesús!


Caleb DRC profile image

Caleb DRC 3 years ago Author

Tomas en Espana,

Yes, the way you wrote it is equivalent to what I have at EL5, and c = 0 in this case. You are correct to include the added constant c.

I will take another look at the text to see if I can clarify it. Thankyou for letting me know it has some problems with clarity.

I believe that Jesus structured calculus into His creation, and also much greater esoteric math to the point that only He understands it; however, you do not need to understand or know the "calculus of Jesus"; you just need to know Jesus to the point you do what He commanded.


Lesa 21 months ago

That's an astute answer to a tricky quesiton


Caleb DRC profile image

Caleb DRC 21 months ago Author

Thankyou Lesa. Frankly, I need to go over all my articles to make them more understandable. Mathematics has been a conduit( used by God) to give us enormous prosperity, but without Christ, what good is it? When it is all said and done, and the eternal outlook is in mind, what good is anything without Christ?

    Sign in or sign up and post using a HubPages Network account.

    0 of 8192 characters used
    Post Comment

    No HTML is allowed in comments, but URLs will be hyperlinked. Comments are not for promoting your articles or other sites.


    Click to Rate This Article
    working