Is there a general way to solve integrals?

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SUMMARY

The discussion centers on the relationship between derivatives and integrals, specifically exploring whether a general formula exists for solving integrals akin to the derivative formula. Participants highlight Riemann sums as a foundational method for calculating integrals, emphasizing that this approach involves taking the limit of the areas of rectangles under a curve as the number of rectangles approaches infinity. The conversation also references the Fundamental Theorem of Calculus (FTC) and questions whether most integrals, particularly those classified as ILATE functions, can be solved using Riemann sums.

PREREQUISITES
  • Understanding of calculus concepts, specifically derivatives and integrals.
  • Familiarity with Riemann sums and their application in integral calculus.
  • Knowledge of the Fundamental Theorem of Calculus (FTC).
  • Basic understanding of ILATE functions in integration.
NEXT STEPS
  • Research the application of Riemann sums in calculating definite integrals.
  • Study the Fundamental Theorem of Calculus and its implications for integral evaluation.
  • Explore the classification and properties of ILATE functions in integration.
  • Investigate alternative methods for solving integrals beyond Riemann sums and the FTC.
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Students and educators in calculus, mathematicians interested in integral calculus, and anyone seeking to deepen their understanding of the relationship between derivatives and integrals.

Voivode
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Homework Statement



This is just something I've been wondering, but since derivatives have the formula:

dy/dx = lim h-> 0 of (f(x+h) - f(x)/h)

And that formula can prove a lot of derivatives.

Does a similar formula exist that can prove integrals?

Homework Equations



The Attempt at a Solution

 
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Yes. Check out riemann sums.

You take the limits of areas of rectangles under the curve as the number of rectangles approaches infinity.
 
CalculusHelp1 said:
Yes. Check out riemann sums.

You take the limits of areas of rectangles under the curve as the number of rectangles approaches infinity.

This is actually the original way that integrals were computed before the discovery of the Fundamental Theorem of Calculus.
 
Would it be possible to prove most integrals without the FTC using Riemann Sums?
 
Voivode said:
Would it be possible to prove most integrals without the FTC using Riemann Sums?

Well, first you'd have to define what you mean by "most integrals". After all, there are an infinite amount of integrals with a solution in the elementary functions. I am relatively certain that all polynomial integrals can be solved in this fashion.
 
By most integrals, I was mainly thinking of the ILATE functions. Could those be solved with riemann sums?
 

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