How much do you memorize to do general derivatives/integrals?

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In summary, you should memorize the following:-monomials of x (if power -1, logarithm)-Qaudratics under root sign: use trig subs-Rational functions of polynomials in trig functions: Tan(pi/2)-substitutions-And exponentials.-Fiddling with product rule and substitution rule to get from 1 to 4, or give up.
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How much do you memorize to do general derivatives/integrals? While it is possible to do everything by going back to first principles, I imagine it gets exhausting to do so every time. So which derivatives/integrals do you memorize? Are the ones that I have attached to this post good enough to have memorized?
 

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  • #2
Turion said:
How much do you memorize to do general derivatives/integrals?

You aren't describing a specific goal. Are you asking how much a student in second semester calculus should memorize? He would also need to know something about exponential functions.
 
  • #3
Well, I haven't thought of it, here's my typical rule
1. monomials of x (if power -1, logarithm)
1b). If in fractions, use fractional decomposition, logaritms for linear factors, arctan for irreducible squares.
2. Qaudratics under root sign: use trig subs,
3. Rational functions of polynomials in trig functions: Tan(pi/2)-substitutions
4. And exponentials.
5. Fiddling with product rule and substitution rule to get from 1 to 4, or give up.
That's about it, I think.
 
  • #4
Stephen Tashi said:
You aren't describing a specific goal. Are you asking how much a student in second semester calculus should memorize? He would also need to know something about exponential functions.

I finished Calculus 1 and 2 and I'm currently taking a course in Differential Equations. I memorized the attachment above for Calculus 2 as advised by the textbook but I find that I have forgotten them over the break. It seems that I still need to have them memorized in order to do integration in Differential Equations but I want to get a general feel of how much I should re-memorize because I don't want to memorize more than I need to.
 
  • #5
arildno said:
Well, I haven't thought of it, here's my typical rule
1. monomials of x (if power -1, logarithm)
1b). If in fractions, use fractional decomposition, logaritms for linear factors, arctan for irreducible squares.
2. Qaudratics under root sign: use trig subs,
3. Rational functions of polynomials in trig functions: Tan(pi/2)-substitutions
4. And exponentials.
5. Fiddling with product rule and substitution rule to get from 1 to 4, or give up.
That's about it, I think.

I see that you have a general algorithm for integration, but do you memorize any derivatives or integrals? For example: you probably have memorized that the derivative of sinx is cosx or that that the derivative of lnx is 1/x.
 
  • #6
basically all you need to know to do derivatives is x' = 1, linearity, and the product and chain rules and the fundamental theorem of calculus. all the derivatives of all the exponential and log and sin and cosine functions and rational functions follow from those. of course you should really memorize the basic ones and not derive them every time.

antiderivatives are harder but again the main tools are linearity, the product rule (integration by parts) and the chain rule (integration by substitution).

To simplify some special integrals it helps to remember your trig identities.

however since only a few special integrals occur as derivatives of familiar functions, antidifferentiation is often of no use in real applied problems, so it is very important to also know how to approximate integrals using monotonicity, and to use power series.
 

1. How much do you need to memorize to do general derivatives and integrals?

As a scientist, I have a strong understanding of calculus and the mathematical principles behind derivatives and integrals. While some basic formulas and rules are helpful to know, the key to success in solving these problems is understanding the underlying concepts and being able to apply them appropriately.

2. Do you need to memorize all the derivative and integral rules?

No, it is not necessary to memorize every single rule for derivatives and integrals. However, having a strong grasp on the fundamental principles and being able to apply them in different scenarios is crucial for success in solving these problems.

3. How important is it to memorize trigonometric identities for calculus?

Trigonometric identities are important to know for calculus, but they can also be derived using basic algebraic principles. It is helpful to have a basic understanding of these identities, but they do not need to be memorized as they can be derived when needed.

4. Can I use a calculator to do derivatives and integrals?

While a calculator can be helpful in checking your work, it is not recommended to solely rely on a calculator for solving derivatives and integrals. It is important to have a strong understanding of the concepts and be able to solve problems without the aid of a calculator.

5. How much practice do I need to do to become proficient in solving derivatives and integrals?

Practice is key in becoming proficient in solving derivatives and integrals. It is important to not only practice solving problems, but also to understand the underlying concepts and principles. With consistent practice and a strong understanding of the fundamentals, you can become proficient in solving these types of problems.

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