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- Thread starter thharrimw
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In summary, thharrimw takes calculus 1 and aptitude test and has trouble with integrations. He wants to know what are the best integrations to use for different questions on the test. He mentions that substitution, integration by parts, and trigonometric substitution are the best methods.f

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- #2

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Hi thharrimw!

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.

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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.

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- #5

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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.

(still thinking about [(tanX)^3]ln(cosX))

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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.

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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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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- #8

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for e^θ/√(1 - e^θ) use U=1-e^θ and du= -e^θ?

Yup!

(4X^3-7X^2+6X-3)/(X^6-4X^3)

urgh … you need to learn LaTeX!

[tex]\frac{4x^3-7x^2+6x-3}{x^6-4x^3}[/tex]

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

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

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

and then I think use partial fractions in the form

[tex]\frac{A}{x}\,+\,\frac{B}{x^2}\,+\,\frac{C}{x^3}\,+\,...[/tex]

How far have you got with the partial fractions?

all of that divided by 96X^2

erm … I couldn't read it anyway …

- #9

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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 [Broken]

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 [Broken]

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- #10

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problems like tan³(X)ln(cosX)

Hint: What is d/dx(sec²x lncosx)?

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What is LaTex and how can i use it

[tex]\frac{4x^3-7x^2+6x-3}{x^6-4x^3}[/tex]

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

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

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

How far have you got with the partial fractions?

i started it but couldn't simplify it

- #12

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What is LaTex and how can i use it

LaTeX is putting [noparse][tex] and [/tex][/noparse] round equations, and they come out

Some people even use LaTeX to build matrices. :tongue2:

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

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

Or just click the "QUOTE" button under other people's posts, and steal

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)

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ok so you +-16 to simplify

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