# Ways to intergrate

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?

tiny-tim
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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?

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. 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 lets us reverse the product rule.

After that I like trigonometric substitution, which lets 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)

tiny-tim
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Hi thharrimw! 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)) 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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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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tiny-tim
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Hi thharrimw! 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! $$\frac{4x^3-7x^2+6x-3}{x^6-4x^3}$$

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

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

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

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? all of that divided by 96X^2

erm … I couldn't read it anyway … mathwonk
Homework Helper
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:

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tiny-tim
Homework Helper
problems like tan³(X)ln(cosX)

Hint: What is d/dx(sec²x lncosx)? urgh … you need to learn LaTeX! 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? i started it but couldn't simplify it

tiny-tim
Homework Helper
What is LaTex and how can i use it

LaTeX is putting [noparse]$$and$$[/noparse] round equations, and they come out really prettily! 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 their LaTeX ideas … that's what I do! 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) Last edited by a moderator:
ok so you +-16 to simplify