Prove of the complicated integral

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In summary, to prove a complicated integral, one must use various mathematical techniques such as substitution, integration by parts, or partial fractions while also having a thorough understanding of the fundamental theorem of calculus. Common mistakes to avoid include mixing up the limits of integration, making a sign error, or forgetting to include a constant of integration. While a complicated integral cannot be solved using only basic calculus techniques, it is possible to simplify it before applying integration techniques. Proving a complicated integral is important for accurately calculating the area under a curve and solving real-world problems, as well as developing problem-solving skills. To improve in this skill, regular practice and a strong understanding of fundamental concepts are key, and seeking help from a tutor or study group can also be beneficial
  • #1
shayaan_musta
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Hi experts!

How to prove this integral?

[itex]\frac{2}{e^{5}}[/itex][itex]\leq[/itex][itex]\int[/itex][itex]\int_{D}[/itex]e[itex]^{-(x^{2}+y^{2})}[/itex]dxdy[itex]\leq[/itex]2
on
D=[0,1] and [0,2]

Here D is subscript of the 2nd integral.

I seriously have no idea how to start. I am 100% blank with this question.

Thanks in advance.
 
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  • #2
What is the maximum/minimum value that [itex]e^{-(x^2+y^2)}[/itex] can attain??
 
  • #3
Why not just calculate the integral? The rectangle (0,1) x (0,2) is diffeomorphic to a circle (or the circle minus a set of measure zero) so use the change of variables theorem.
 

FAQ: Prove of the complicated integral

How do you prove a complicated integral?

To prove a complicated integral, you must use various mathematical techniques such as substitution, integration by parts, or partial fractions. You must also have a thorough understanding of the fundamental theorem of calculus and be able to apply it correctly.

What are some common mistakes to avoid when proving a complicated integral?

Some common mistakes to avoid when proving a complicated integral include mixing up the limits of integration, making a sign error, or forgetting to include a constant of integration. It is also important to carefully check your work for any algebraic mistakes.

Can a complicated integral be solved using only basic calculus techniques?

In most cases, a complicated integral cannot be solved using only basic calculus techniques. However, it is possible to simplify the integral by using algebraic manipulation or trigonometric identities before applying integration techniques.

Why is it important to prove a complicated integral?

Proving a complicated integral is important because it allows us to accurately calculate the area under a curve and solve various real-world problems. It also helps us understand the underlying principles of calculus and develop our problem-solving skills.

How can I improve my skills in proving complicated integrals?

To improve your skills in proving complicated integrals, it is essential to practice regularly and to have a strong understanding of the fundamental concepts of calculus. You can also seek help from a tutor or join a study group to discuss and solve challenging problems together.

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