- #1
mathslover
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How should we proceed to find the definite integral
Int[ log(sin(x))*log(cos(x)) ,{x,0,pi/2} ] ?
mathslover
Int[ log(sin(x))*log(cos(x)) ,{x,0,pi/2} ] ?
mathslover
How Do We Solve the Definite Integral of log(sin(x))*log(cos(x)) from 0 to pi/2?
- Thread starter mathslover
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AnhIn summary, to find the definite integral Int[ log(sin(x))*log(cos(x)) ,{x,0,pi/2} ] , you can express cos/sin in terms of an infinite product, split up the logs, and then continue with the solution. This method was also used by Wolstenholme in his solution to a similar problem.
- #2
Gib Z
Homework Helper
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I can't remember the source of even the exact answer :( But I've seen integrals of that form done in some papers by Ramanujan where he expressed the answer in terms of the zeta function and something else, so it's not exactly an elementary integral.
This isn't how he did it, and I'm not sure if it will help, but you could express cos/sin in terms of an infinite product, split up the logs ( log [ab]= log [a] + log ) and then continue?
This isn't how he did it, and I'm not sure if it will help, but you could express cos/sin in terms of an infinite product, split up the logs ( log [ab]= log [a] + log ) and then continue?
- #3
1. What is a difficult integral?
A difficult integral is a type of mathematical problem that involves finding the area under a curve (or the definite integral) using complex equations and techniques. These integrals often require advanced knowledge of calculus and special methods to solve.
2. Why are difficult integrals important?
Difficult integrals are important in various fields of science, engineering, and mathematics. They are used to solve complex problems, model real-world situations, and make accurate predictions. They also help us understand the behavior of functions and their relationships with other variables.
3. What are some strategies for solving difficult integrals?
Some strategies for solving difficult integrals include using substitution, integration by parts, trigonometric identities, and special techniques such as partial fractions and series expansion. It is also important to have a good understanding of the fundamental properties of integrals and theorems such as the Fundamental Theorem of Calculus.
4. How can I improve my skills in solving difficult integrals?
Practice is key to improving your skills in solving difficult integrals. Make sure to have a strong foundation in calculus and constantly challenge yourself with new problems. It can also be helpful to work with others, seek guidance from a mentor or teacher, and use online resources and textbooks.
5. Are there any real-world applications of difficult integrals?
Yes, difficult integrals have many real-world applications in fields such as physics, engineering, economics, and statistics. They are used to calculate areas, volumes, work, and probabilities, among other things. For example, they are used in modeling fluid flow, determining the stability of structures, and predicting the behavior of financial markets.
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