Difficult integral (for me)

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B5ZWFyIHN1bW1hcml6ZXIgY2FuIGxldCB5IHVzZSB1bnRpbCB5b3Ugc2V0IHlvdXJzLg==In summary, The integral \int_0^1 \sqrt[3]{1-x^7}-\sqrt[7]{1-x^3} dx can be solved by using the substitution u=\sqrt[3]{1-x^7} and u=\sqrt[7]{1-x^3}, and then considering the geometry of the integral in terms of the relation y^3 + x^7 = 1. Alternatively, in the first integral, letting y =
  • #1

Homework Statement



[itex]\int_0^1 \sqrt[3]{1-x^7}-\sqrt[7]{1-x^3} dx[/itex]

Homework Equations



None

The Attempt at a Solution



I tried using the substitutions [itex]u=\sqrt[3]{1-x^7}[/itex] and [itex]u=\sqrt[7]{1-x^3}[/itex] to no avail and I couldn't think of any more substitutions. Any suggestions?
 
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  • #2
express the roots as fractional powers
treat the two terms separately

consider the geometry of the integral in terms of the relation
[itex]y^3 + x^7 = 1[/itex]

the roles of x and y swap in the second integral don't they?
 
  • #3
blockcolder said:

Homework Statement



[itex]\int_0^1 \sqrt[3]{1-x^7}-\sqrt[7]{1-x^3} dx[/itex]

Homework Equations



None

The Attempt at a Solution



I tried using the substitutions [itex]u=\sqrt[3]{1-x^7}[/itex] and [itex]u=\sqrt[7]{1-x^3}[/itex] to no avail and I couldn't think of any more substitutions. Any suggestions?

In the first integral, let y = x^7. Then look up "Beta function".

RGV
 

What is an integral?

An integral is a mathematical concept that represents the area under a curve on a graph. It is used to calculate the total value of a quantity that is changing continuously over a given interval.

Why are some integrals considered difficult?

Some integrals are considered difficult because they cannot be solved using simple algebraic methods. They require advanced techniques and strategies to solve, such as integration by parts, substitution, or trigonometric identities.

How can I approach a difficult integral?

There are several strategies that can help you approach a difficult integral. These include breaking the integral into smaller parts, using substitution to simplify the integrand, and looking for patterns or symmetries in the integral.

What are some common mistakes to avoid when solving a difficult integral?

Common mistakes when solving a difficult integral include forgetting to use the correct integration technique, making algebraic errors, and forgetting to include the constant of integration in the final answer.

What resources are available to help me solve a difficult integral?

There are many online resources available, such as integral calculators, step-by-step guides, and instructional videos, that can help you solve difficult integrals. You can also consult textbooks, ask your peers or instructor for help, or seek out a tutor for additional support.

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