Integration Homework: Substitution, Partial Fraction, By Parts

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In summary, the integral provided in the conversation cannot be solved using only elementary methods and may require the use of non-elementary functions or an infinite series of elementary functions.
  • #1
kishlaysingh
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Homework Statement


integrate
integral.JPG


Homework Equations


please solve this using methods only like
1. Substitution
2.Partial fraction
3.By Parts

The Attempt at a Solution


i have tried all the above three methods mainly using substitution and by parts...
i have expanded the a^3 - x^3 and then kept the a/2 - x as "p" but still the function doesn't simplify
 
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  • #2
What context did this appear in? It doesn't appear to have an elementary antiderivative.
 
  • #3
kishlaysingh said:

Homework Statement


integrate
View attachment 80770

Homework Equations


please solve this using methods only like
1. Substitution
2.Partial fraction
3.By Parts

The Attempt at a Solution


i have tried all the above three methods mainly using substitution and by parts...
i have expanded the a^3 - x^3 and then kept the a/2 - x as "p" but still the function doesn't simplify

The integral is "non-elementary", so cannot be done in terms of a finite expression involving only elementary functions (such as powers, roots, trigonometric functions and their inverses, exponentials and logarithms, etc.). However, it can be expressed in terms of the non-elementary "elliptic functions", or it can be expressed in terms of an infinite series of elementary functions.

Note: the non-elementary nature of the integral is a theorem that has been proven rigorously. It is not just a matter of somebody not being smart enough to figure out how to do the integral (in finitely many terms and factors of elementary functions); rather, it has been proven to be impossible. So, even if you allow yourself to use a formula taking 10 billion pages to write out in detail, it still would not be the answer to your question!
 
  • #4
ok, fine then...:oldcry:
 

What is the purpose of integration homework?

The purpose of integration homework is to help students practice and develop their skills in solving integration problems using various techniques such as substitution, partial fraction, and integration by parts.

What is substitution in integration?

Substitution is a technique used in integration to simplify a complex integral by replacing the variable with a new variable or expression. This allows the integral to be rewritten in a form that is easier to solve.

What is partial fraction in integration?

Partial fraction is a technique used to decompose a rational function into simpler fractions, which can then be integrated separately. This is useful for solving integrals involving rational functions.

What is integration by parts?

Integration by parts is a technique used to integrate the product of two functions. It involves breaking down the integral into two parts and applying the product rule of differentiation in reverse to solve for the integral.

What are some tips for solving integration problems?

Some tips for solving integration problems include identifying the correct technique to use, being familiar with basic integration rules and formulas, carefully choosing the substitution or partial fractions, and practicing regularly to improve problem-solving skills.

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