Solve Calculus Integration Problems: Tips & Advice

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Discussion Overview

The discussion revolves around various methods and strategies for solving calculus integration problems, particularly focusing on techniques that can be useful for students preparing for exams. Participants share their experiences with specific integration challenges and seek advice on effective approaches.

Discussion Character

  • Exploratory
  • Technical explanation
  • Homework-related
  • Debate/contested

Main Points Raised

  • Some participants suggest that learning substitution is foundational for integration, followed by integration by parts and trigonometric substitution.
  • One participant expresses uncertainty about how to handle specific integrals, such as tan³(X)ln(cosX), and seeks guidance on the best methods to apply.
  • Another participant lists various methods they are familiar with, including separation, long division, inverse trigonometric functions, and partial fractions, and asks if there are additional methods they should learn.
  • There are hints provided for specific integrals, suggesting the use of partial fractions and substitutions for certain expressions.
  • One participant argues that there are fundamentally only two methods of integration: substitution and integration by parts, while noting that other techniques apply to specific function types.
  • Discussion includes a mention of advanced topics such as elliptic functions and the classification of functions that cannot be integrated using elementary functions.
  • Participants discuss the use of LaTeX for formatting mathematical expressions in their posts, with some expressing confusion about its application.

Areas of Agreement / Disagreement

Participants do not reach a consensus on the best methods for integration, and multiple competing views on the importance and utility of various techniques remain. There is also uncertainty about the effectiveness of certain approaches for specific problems.

Contextual Notes

Some participants express limitations in their knowledge due to gaps in their education or teaching experience, which may affect their understanding of integration techniques. Additionally, there are unresolved mathematical steps and assumptions regarding the applicability of certain methods to specific integrals.

Who May Find This Useful

This discussion may be useful for students studying calculus, particularly those preparing for exams, as well as educators looking for insights into common challenges faced by learners in integration techniques.

thharrimw
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I have tought myself calculus 1 and took the ap calc test. on the test i had problems woth the intergratinos so i was wondering if anyone could tell me what are the most usefull ways to intergrate?
 
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thharrimw said:
I have tought myself calculus 1 and took the ap calc test. on the test i had problems woth the intergratinos so i was wondering if anyone could tell me what are the most usefull ways to intergrate?

Hi thharrimw! :smile:

I take it from your question that you know all the ways, but you don't know how to choose the best one for each particular question?

So you start doing a question one way, and then find it doesn't work, and you've wasted exam time and you have to start again?

Show us which integrations you had problems with. :smile:
 
I think that the best way to build a good foundation for integration (anti-derivatives) is to learn substitution, which allows us to reverse the chain rule.

After that my next favorite method is integration by parts, which let's us reverse the product rule.

After that I like trigonometric substitution, which let's us reverse derivatives of inverse trig functions like the arcsine.
 
problems like tan(X)^3ln(cosX)- integration by parts, but I don't klnow what to do with the ^3. The methods I know of are: Separation, Long Division, Inverse trig, Completing the Square, Partial Fractions , u substution ,and Integration by Parts. I am working on how to do Trig integrations. Are there any more methods I need to learn? What are the most usefull? I don't have a teacher so I have been just treating everything like it's really needed and learning how to do it by doing at least 50-60 problens of that type. here are some more problems I don't Know how to intergrate, 1/(T^(1/2)(T^(1/2)+1), (2^(lnx))/x, e^(Theta)/(1-e^(Theta)0^(1/2)
 
Hi thharrimw! :smile:

Hints: For 1/(√T)(1 + √T), use partial fractions and u-substitution (in either order).

For (2^(lnx))/x, use 2 = e^(log2).

For e^θ/√(1 - e^θ), use the obvious substitution. :smile:

(still thinking about [(tanX)^3]ln(cosX)) :frown:
 
for e^θ/√(1 - e^θ) use U=1-e^θ and du= -e^θ?

i found this problem and i did it with my TI-89 but how wolud you find the integral of (4X^3-7X^2+6X-3)/(X^6-4X^3) by hand (my calc gave me -[2(13*2^(1/3)-12)(√3*2^(1/3))X^2*Arctan[√3-*2^(1/3)(2X+2^(2/3))/6]
+2^(2/3)+28√2+13)X^2ln(X62+2^2/3X+X^(4/3)+2(2^(1/3)(14*2^(2/3)*X^2ln|X-2^(2/3)|-6(14X^2ln|X|)+2(4X-1)))))]
i tryed to use partial fractions but i couldn't solve it using partial fractions so i have no idea of where to start.
 
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thharrimw said:
for e^θ/√(1 - e^θ) use U=1-e^θ and du= -e^θ?

i found this problem and i did it with my TI-89 but how wolud you find the integral of (4X^3-7X^2+6X-3)/(X^6-4X^3) by hand (my calc gave me -[2(13*2^(1/3)-12)(√3*2^(1/3))X^2*Arctan[√3-*2^(1/3)(2X+2^(2/3))/6]
+2^(2/3)+28√2+13)X^2ln(X62+2^2/3X+X^(4/3)+2(2^(1/3)(14*2^(2/3)*X^2ln|X-2^(2/3)|-6(14X^2ln|X|)+2(4X-1)))))]
i tryed to use partial fractions but i couldn't solve it using partial fractions so i have no idea of where to start.

all of that divided by 96X^2
 
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Hi thharrimw! :smile:
thharrimw said:
for e^θ/√(1 - e^θ) use U=1-e^θ and du= -e^θ?

Yup! :smile:
(4X^3-7X^2+6X-3)/(X^6-4X^3)

urgh … you need to learn LaTeX! :eek:

\frac{4x^3-7x^2+6x-3}{x^6-4x^3}

=\,\frac{4x^3-7x^2+6x-3}{x^3(x^3-4)}

=\,\frac{4}{x^3}\,+\,\frac{-7x^2+6x+13}{x^3(x^3-4)},

(the quadratic is (x+1)(7x-13), but I don't think that helps :cry:)

and then I think use partial fractions in the form

\frac{A}{x}\,+\,\frac{B}{x^2}\,+\,\frac{C}{x^3}\,+\,...

How far have you got with the partial fractions? :smile:​
thharrimw said:
all of that divided by 96X^2

erm … I couldn't read it anyway … :wink:
 
there are only two methods of integration, substitution, and "parts".

all other tricks concern particular types of functions, like trig funcs, and rational functions (partial fractions).

my school courses skipped these things, or i skipped those classes. the resulting gap in knowledge was never relevant to my research career, but only to my teaching career.

i.e. these things are taught but almost never used. hence after many years teaching them, i at last became familiar with them, but still never used them, except in teaching other courses like diff eq and several variables calc.

the integrals most interesting in research, are those which CANNOT be anti - differentiated by elementary functions,

which lead to elliptic functions and other esoteric concepts like the jacobian of an algebraic curve.

the theory of classifying exactly which functions can be anti differentiated by elementary terms, is however quite interesting, and has been essentially perfected, so that the process is no longer truly considered an art. i.e. there are actually algorithms which work when possible, and tell when the job is hopeless.

names like rosenlicht, ritt, and more recent ones occur here. you may search for articles on "integration in elementary terms".

here is a nice survey:

http://www.claymath.org/programs/outreach/academy/LectureNotes05/Conrad.pdf
 
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  • #10
thharrimw said:
problems like tan³(X)ln(cosX)

Hint: What is d/dx(sec²x lncosx)? :smile:
 
  • #11
urgh … you need to learn LaTeX! :eek:

What is LaTex and how can i use it


\frac{4x^3-7x^2+6x-3}{x^6-4x^3}

=\,\frac{4x^3-7x^2+6x-3}{x^3(x^3-4)}

=\,\frac{4}{x^3}\,+\,\frac{-7x^2+6x+13}{x^3(x^3-4)},

how can you pull 4/X^3 out of the problem.



How far have you got with the partial fractions? :smile:​


i started it but couldn't simplify it
 
  • #12
thharrimw said:
What is LaTex and how can i use it


LaTeX is putting [noparse]and[/noparse] round equations, and they come out really prettily! :smile:

Some people even use LaTeX to build matrices. :-p

There's a sticky thread somwhere on this forum explaining all about LaTeX … but I can't find it! :redface:

Look at, and bookmark http://www.physics.udel.edu/~dubois/lshort2e/node61.html#SECTION008100000000000000000 and http://www.physics.udel.edu/~dubois/lshort2e/node54.html#SECTION00830000000000000000 :smile:

Or just click the "QUOTE" button under other people's posts, and steal their LaTeX ideas … that's what I do! :rolleyes:

btw, your "quotes" from other posts look odd … are you using an antiquated computer system? Do you get lots of symbols at the top of your Reply to Thread box (like a drop-down list of smilies)? If so, the ∑ button will help with the LaTeX.
how can you pull 4/X^3 out of the problem.

'cos 4/x³ = (4x³ - 16)/x³(x³ - 4) :smile:
 
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  • #13
ok so you +-16 to simplify
 

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